id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-9400 | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0 | Let $c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | First:
{{begin-eqn}}
{{eqn | l = c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + b x}
| c =
}}
{{eqn | r = \int \frac {\d x} {x \paren {a x + b} }
| c =
}}
{{eqn | r = \frac 1 b \ln \size {\frac x {a x + b} } + C
... | Let $c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | First:
{{begin-eqn}}
{{eqn | l = c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + b x}
| c =
}}
{{eqn | r = \int \frac {\d x} {x \paren {a x + b} }
| c =
}}
{{eqn | r = \frac 1 b \ln \size {\frac x {a x + b} } + C
... | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0/Proof_1 | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of x by a x + b"
] |
proofwiki-9401 | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0 | Let $c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | Let $c = 0$.
From Primitive of $\dfrac 1 {a x^2 + b x + c}$, we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - ... | Let $c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {\frac x {a x + b} } + C$ | Let $c = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus c|Primitive of $\dfrac 1 {a x^2 + b x + c}$]], we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\df... | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0/Proof_2 | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9402 | Primitive of x squared over a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a x^2 + b x + c}
| r = \int \frac 1 a \paren {1 - \frac {b x + c} {a x^2 + b x + c} } \rd x
| c = by division
}}
{{eqn | r = \frac 1 a \int \rd x - \frac b a \int \frac {x \rd x} {a x^2 + b x + c} - \frac c a \int \frac {\d x} {a x^2 + b x + c}
| c = L... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {a x^2 + b x + c}
| r = \int \frac 1 a \paren {1 - \frac {b x + c} {a x^2 + b x + c} } \rd x
| c = by division
}}
{{eqn | r = \frac 1 a \int \rd x - \frac b a \int \frac {x \rd x} {a x^2 + b x + c} - \frac c a \int \frac {\d x} {a x^2 + b x + c}
| c = [... | Primitive of x squared over a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c | [
"Primitive of x squared over a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of x over a x squared plus b x plus c"
] |
proofwiki-9403 | Primitive of x squared over a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x
| r = \int \dfrac {6 x^2 + 8 x + 4} {3 x^2 + 4 x + 2} \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x
| c = Linear Combination of Primitives
}}
{{eqn | r = \int 2 \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x
... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x
| r = \int \dfrac {6 x^2 + 8 x + 4} {3 x^2 + 4 x + 2} \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int 2 \rd x + \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x
... | Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c/Examples/6_x^2_+_10_x_+_5_over_3_x^2_+_4_x_+_2/Proof_1 | [
"Primitive of x squared over a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of p x + q over a x squared plus 2 b x plus c/Examples/2 x + 1 over 3 x^2 + 4 x + 2"
] |
proofwiki-9404 | Primitive of x squared over a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x
| r = 6 \int \dfrac {x^2} {3 x^2 + 4 x + 2} \rd x + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + 5 \int \dfrac {\d x} {3 x^2 + 4 x + 2}
| c = Linear Combination of Primitives
}}
{{eqn | r = 6 \paren {\frac x 3 - \frac 4 {2 \times... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {a x^2 + b x + c} = \frac x a - \frac b {2 a^2} \ln \size {a x^2 + b x + c} + \frac {b^2 - 2 a c} {2 a^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \dfrac {6 x^2 + 10 x + 5} {3 x^2 + 4 x + 2} \rd x
| r = 6 \int \dfrac {x^2} {3 x^2 + 4 x + 2} \rd x + 10 \int \dfrac x {3 x^2 + 4 x + 2} \rd x + 5 \int \dfrac {\d x} {3 x^2 + 4 x + 2}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = 6 \paren {\frac x 3 - \frac 4 {2 \t... | Primitive of x squared over a x squared plus b x plus c/Examples/6 x^2 + 10 x + 5 over 3 x^2 + 4 x + 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_squared_plus_b_x_plus_c/Examples/6_x^2_+_10_x_+_5_over_3_x^2_+_4_x_+_2/Proof_2 | [
"Primitive of x squared over a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x squared over a x squared plus b x plus c",
"Primitive of x over a x squared plus b x plus c",
"Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2"
] |
proofwiki-9405 | Primitive of Power of x over a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^m \rd x} {a x^2 + b x + c} = \frac {x^{m - 1} } {\paren {m - 1} a} - \frac b a \int \frac {x^{m - 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{m - 2} \rd x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {a x^2 + b x + c}
| r = \int \frac 1 a \frac {a x^m \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} a x^2 \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} \paren {a x^2 + b x + c - b x - c} \r... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^m \rd x} {a x^2 + b x + c} = \frac {x^{m - 1} } {\paren {m - 1} a} - \frac b a \int \frac {x^{m - 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{m - 2} \rd x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {a x^2 + b x + c}
| r = \int \frac 1 a \frac {a x^m \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} a x^2 \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{m - 2} \paren {a x^2 + b x + c - b x - c} \r... | Primitive of Power of x over a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9406 | Primitive of Reciprocal of a x squared plus b x plus c/Negative Discriminant | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c < 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$ | Let $b^2 - 4 a c < 0$.
Then:
{{begin-eqn}}
{{eqn | l = - D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = - D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-... | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c < 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$ | Let $b^2 - 4 a c < 0$.
Then:
{{begin-eqn}}
{{eqn | l = - D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = - D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {4 a c - b^2}
| c = by definition of $D$
}}
{{end-eqn}}
Thus:
{{b... | Primitive of Reciprocal of a x squared plus b x plus c/Negative Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Negative_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Negative_Discriminant | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9407 | Primitive of Reciprocal of a x squared plus b x plus c/Positive Discriminant | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c > 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = Definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{... | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c > 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C$ | Let $b^2 - 4 a c > 0$.
Then:
{{begin-eqn}}
{{eqn | l = D
| o = >
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = D
| r = q^2
| c = for some $q \in \R$
}}
{{eqn | ll= \leadsto
| l = q
| r = \sqrt {b^2 - 4 a c}
| c = Definition of $D$
}}
{{end-eqn}}
Thus:
{{begin-eq... | Primitive of Reciprocal of a x squared plus b x plus c/Positive Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Positive_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Positive_Discriminant | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9408 | Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac {-2} {2 a x + b} + C$ | Let $b^2 - 4 a c = 0$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {4 a \rd x} {\paren {2 a x + b}^2}
| c = from $(1)$
}}
{{eqn | r = \frac {-4 a} {2 a \paren {2 a x + b} } + C
| c = Primitive of $\dfrac 1 {\paren {a x + b}^2}$
}}
{{eqn | r = \dfrac {-2} {2 a x... | Let $a \in \R_{\ne 0}$.
Let $b^2 - 4 a c = 0$.
Then:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac {-2} {2 a x + b} + C$ | Let $b^2 - 4 a c = 0$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {4 a \rd x} {\paren {2 a x + b}^2}
| c = from $(1)$
}}
{{eqn | r = \frac {-4 a} {2 a \paren {2 a x + b} } + C
| c = [[Primitive of Reciprocal of a x + b squared|Primitive of $\dfrac 1 {\paren {... | Primitive of Reciprocal of a x squared plus b x plus c/Zero Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Zero_Discriminant | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Zero_Discriminant | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-9409 | Primitive of Reciprocal of x by a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c} } = \frac 1 {2 c} \ln \size {\frac {x^2} {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x^2 + b x + c} }
| r = \int \paren {\frac 1 {c x} - \frac {a x + b} {c \paren {a x^2 + b x + c} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 c \int \frac {\d x} x - \frac a c \int \frac {x \rd x} {a x^2 + b x + c} - \frac b c \int... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c} } = \frac 1 {2 c} \ln \size {\frac {x^2} {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x^2 + b x + c} }
| r = \int \paren {\frac 1 {c x} - \frac {a x + b} {c \paren {a x^2 + b x + c} } } \rd x
| c = [[Primitive of Reciprocal of x by a x squared plus b x plus c/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac ... | Primitive of Reciprocal of x by a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of x by a x squared plus b x plus c/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over a x squared plus b x plus c",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9410 | Primitive of Reciprocal of x squared by a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} } = \frac b {2 c^2} \ln \size {\frac {a x^2 + b x + c} {x^2} } - \frac 1 {c x} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} }
}}
{{eqn | r = \int \paren {\frac {-b} {c^2 x} + \frac 1 {c x^2} + \frac {a b x + b^2 - a c} {c^2 \paren {a x^2 + b x + c} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac {-b} {c^2} \int \frac {\d x} x + \frac... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} } = \frac b {2 c^2} \ln \size {\frac {a x^2 + b x + c} {x^2} } - \frac 1 {c x} + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c} }
}}
{{eqn | r = \int \paren {\frac {-b} {c^2 x} + \frac 1 {c x^2} + \frac {a b x + b^2 - a c} {c^2 \paren {a x^2 + b x + c} } } \rd x
| c = [[Primitive of Reciprocal of x squared by a x squared plus b x plus c/Partial Fraction... | Primitive of Reciprocal of x squared by a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of x squared by a x squared plus b x plus c/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power",
"Primitive of x over a x squared plus b x plus c",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9411 | Primitive of Reciprocal of Power of x by a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^n \paren {a x^2 + b x + c} } = \frac {-1} {\paren {n - 1} c x^{n - 1} } - \frac b c \int \frac {\d x} {x^{n - 1} \paren {a x^2 + b x + c} } - \frac a c \int \frac {\d x} {x^{n - 2} \paren {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^{n - 2} \paren {a x^2 + b x + c} }
| r = \int \frac {x^{-n + 2} \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac {x^{-n + 1} } {\paren {-n + 1} a} - \frac b a \int \frac {x^{-n + 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{-n} \rd x} {a x^2 + b x... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^n \paren {a x^2 + b x + c} } = \frac {-1} {\paren {n - 1} c x^{n - 1} } - \frac b c \int \frac {\d x} {x^{n - 1} \paren {a x^2 + b x + c} } - \frac a c \int \frac {\d x} {x^{n - 2} \paren {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^{n - 2} \paren {a x^2 + b x + c} }
| r = \int \frac {x^{-n + 2} \rd x} {a x^2 + b x + c}
| c =
}}
{{eqn | r = \frac {x^{-n + 1} } {\paren {-n + 1} a} - \frac b a \int \frac {x^{-n + 1} \rd x} {a x^2 + b x + c} - \frac c a \int \frac {x^{-n} \rd x} {a x^2 + b x... | Primitive of Reciprocal of Power of x by a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Primitive of Power of x over a x squared plus b x plus c"
] |
proofwiki-9412 | Primitive of Reciprocal of square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {a x^2 + b x + c}^2} = \frac {2 a x + b} {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 a} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = 2 a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a
| c = Derivative of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
| r = \int \paren {\frac {4 a} {\paren {2 a x + ... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {a x^2 + b x + c}^2} = \frac {2 a x + b} {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 a} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = 2 a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a
| c = [[Derivative of Power]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x^2 + b x + c}^2}
| r = \int \paren {\frac {4 a} {\paren {2 ... | Primitive of Reciprocal of square of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Completing the Square",
"Integration by Substitution",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of p x + q by Root of a x + b",
"Completing the Square"
] |
proofwiki-9413 | Completing the Square | Let $a, b, c, x$ be real numbers with $a \ne 0$.
Then:
:$a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$
This process is known as '''completing the square'''. | {{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {4 a^2 x^2 + 4 a b x + 4 a c} {4 a}
| c = multiplying top and bottom by $4 a$
}}
{{eqn | r = \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}
| c =
}}
{{eqn | r = \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}
| c =
}}
{{end-eqn}}
{{... | Let $a, b, c, x$ be [[Definition:Real Number|real numbers]] with $a \ne 0$.
Then:
:$a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$
This process is known as '''[[Completing the Square|completing the square]]'''. | {{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {4 a^2 x^2 + 4 a b x + 4 a c} {4 a}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $4 a$
}}
{{eqn | r = \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}
| c =
}}
{{eqn | r = \frac {\paren {2 a x + b}^2... | Completing the Square | https://proofwiki.org/wiki/Completing_the_Square | https://proofwiki.org/wiki/Completing_the_Square | [
"Completing the Square",
"Algebra",
"Proof Techniques"
] | [
"Definition:Real Number",
"Completing the Square"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9414 | Primitive of x over square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} = \frac {-\paren {b x + 2 c} } {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac b {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = Derivative of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2}
| c = ... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2} = \frac {-\paren {b x + 2 c} } {\paren {4 a c - b^2} \paren {a x^2 + b x + c} } - \frac b {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = [[Derivative of Power]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {x \rd x} {\paren {a x^2 + b x + c}^2}
... | Primitive of x over square of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_over_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_over_square_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Integration by Substitution",
"Primitive of Power",
"Primitive of Reciprocal of square of a x squared plus b x plus c",
"Definition:Common Denominator"
] |
proofwiki-9415 | Primitive of x squared over square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {a x^2 \rd x} {\paren {a x^2 + b x + c}^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {\p... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2} = \frac {\paren {b^2 - 2 a c} x + b c} {a \paren {4 a c - b^2} \paren {a x^2 + b x + c} } + \frac {2 c} {4 a c - b^2} \int \frac {\d x} {a x^2 + b x + c}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {a x^2 \rd x} {\paren {a x^2 + b x + c}^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 a \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x}... | Primitive of x squared over square of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_square_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over square of a x squared plus b x plus c",
"Primitive of Reciprocal of square of a x squared plus b x plus c"
] |
proofwiki-9416 | Primitive of Power of x over Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
| r = \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }
| c =
}}
{{eqn | o =
| ro= +
| r = \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m -... | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac 1 {\paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-\paren {2 a x + b... | Let $a \in \R_{\ne 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {a x^2 + b x + c}^n}
| r = \frac {x^{m - 1} } {\paren {2 n - m - 1} a \paren {a x^2 + b x + c}^{n - 1} }
| c =
}}
{{eqn | o =
| ro= +
| r = \frac {\paren {m - 1} c} {\paren {2 n - m - 1} a} \int \frac {x^{m ... | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac 1 {\paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-\paren {2 a x + ... | Primitive of Power of x over Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9417 | Primitive of Odd Power of x over Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n} = \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{2 n - 3} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \int \frac {x^{2 n - 3} \paren {a x^2... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n} = \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {x^{2 n - 3} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = \frac 1 a \int \frac {x^{2 n - 3} \paren {a... | Primitive of Odd Power of x over Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Odd_Power_of_x_over_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Odd_Power_of_x_over_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9418 | Primitive of Reciprocal of x by square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c}^2} = \frac 1 {2 c \paren {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}
| r = \int \frac {c \rd x} {c x \paren {a x^2 + b x + c}^2}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \rd x} {x \paren {a x^2 + b x + c}^2}
| c = Primitive of Constant Multiple of Function
}... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c}^2} = \frac 1 {2 c \paren {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^2} + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}
| r = \int \frac {c \rd x} {c x \paren {a x^2 + b x + c}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \rd x} {x \paren {a x^2 + b x + c}^2}
... | Primitive of Reciprocal of x by square of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_square_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function under its Derivative"
] |
proofwiki-9419 | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$ | From Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} a} {... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$ | From [[Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c]]:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} ... | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c/Proof_1 | [
"Primitives involving a x squared plus b x plus c",
"Primitive of Reciprocal of x squared by square of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c"
] |
proofwiki-9420 | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$ | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {a x^2 + b x + c}^2}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \rd x} {x^2 \paren {a x^2 + b x + c}^2}
| c = Pr... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2} = \frac {-1} {c x \paren {a x^2 + b x + c} } - \frac {3 a} c \int \frac {\d x} {\paren {a x^2 + b x + c}^2} - \frac {2 b} c \int \frac {\d x} {x \paren {a x^2 + b x + c}^2}$ | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {a x^2 + b x + c}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \... | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c/Proof_2 | [
"Primitives involving a x squared plus b x plus c",
"Primitive of Reciprocal of x squared by square of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Primitive of Power",
"Integration by Parts",
"Linear Combination of Integ... |
proofwiki-9421 | Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren ... | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^m \paren {a x^2 + b x + c}^n}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \rd x} {x^m \paren {a x^2 + b x + c}^n}
| c = Pr... | Let $a \in \R_{\ne 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} a} {\paren {m - 1} c} \int \frac {\d x} {x^{m - 2} \paren... | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^m \paren {a x^2 + b x + c}^n}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \... | Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Primitive of Power",
"Integration by Parts",
"Linear Combination of Integ... |
proofwiki-9422 | Ring of Sets is Closed under Finite Union | Let $\RR$ be a ring of sets.
Let $A_1, A_2, \ldots, A_n \in \RR$.
Then:
:$\ds \bigcup_{j \mathop = 1}^n A_j \in \RR$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \bigcup_{j \mathop = 1}^n A_j \in \RR$
$\map P 1$ is true, as this just says $A_1 \in \RR$. | Let $\RR$ be a [[Definition:Ring of Sets|ring of sets]].
Let $A_1, A_2, \ldots, A_n \in \RR$.
Then:
:$\ds \bigcup_{j \mathop = 1}^n A_j \in \RR$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \bigcup_{j \mathop = 1}^n A_j \in \RR$
$\map P 1$ is true, as this just says $A_1 \in \RR$. | Ring of Sets is Closed under Finite Union | https://proofwiki.org/wiki/Ring_of_Sets_is_Closed_under_Finite_Union | https://proofwiki.org/wiki/Ring_of_Sets_is_Closed_under_Finite_Union | [
"Rings of Sets"
] | [
"Definition:Ring of Sets"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-9423 | Sigma-Ring contains Limit Superior of Sequence of Sets | Let $\RR$ be a $\sigma$-ring.
Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a sequence of sets in $\RR$.
Then:
:$\ds \limsup_{n \mathop \to \infty} A_n \in \RR$ | Define the sequence of sets:
:$B_i := \ds \bigcup_{n \mathop = i}^\infty A_n$
From $\sigma$-ring axiom $(\text {SR} 3)$: Closure under Countable Unions:
:$\forall i: B_i \in \RR$
From Sigma-Ring is Closed under Countable Intersections:
:$\forall n: \ds \bigcap_{i \mathop = 0}^\infty B_i \in \RR$
Thus:
{{begin-eqn}}
{{e... | Let $\RR$ be a [[Definition:Sigma-Ring|$\sigma$-ring]].
Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in $\RR$.
Then:
:$\ds \limsup_{n \mathop \to \infty} A_n \in \RR$ | Define the [[Definition:Sequence|sequence]] of [[Definition:Set|sets]]:
:$B_i := \ds \bigcup_{n \mathop = i}^\infty A_n$
From [[Axiom:Sigma-Ring Axioms/Formulation 1|$\sigma$-ring axiom $(\text {SR} 3)$]]: [[Definition:Closed Algebraic Structure|Closure]] under [[Definition:Countable|Countable]] [[Definition:Set Unio... | Sigma-Ring contains Limit Superior of Sequence of Sets | https://proofwiki.org/wiki/Sigma-Ring_contains_Limit_Superior_of_Sequence_of_Sets | https://proofwiki.org/wiki/Sigma-Ring_contains_Limit_Superior_of_Sequence_of_Sets | [
"Sigma-Rings"
] | [
"Definition:Sigma-Ring",
"Definition:Sequence",
"Definition:Set"
] | [
"Definition:Sequence",
"Definition:Set",
"Axiom:Sigma-Ring Axioms/Formulation 1",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Countable Set",
"Definition:Set Union",
"Sigma-Ring is Closed under Countable Intersections"
] |
proofwiki-9424 | Gamma Function for Non-Negative Integer Argument | The Gamma function satisfies:
:$\map \Gamma z = \dfrac {\map \Gamma {z + 1} } z$
for any $z$ which is not a nonpositive integer. | From Gamma Difference Equation:
:$\map \Gamma {z + 1} = z \, \map \Gamma z$
which is valid for all $z \notin \Z_{\le 0}$.
The result follows by dividing by $z$. | The [[Definition:Gamma Function|Gamma function]] satisfies:
:$\map \Gamma z = \dfrac {\map \Gamma {z + 1} } z$
for any $z$ which is not a [[Definition:Negative Integer|nonpositive integer]]. | From [[Gamma Difference Equation]]:
:$\map \Gamma {z + 1} = z \, \map \Gamma z$
which is valid for all $z \notin \Z_{\le 0}$.
The result follows by dividing by $z$. | Gamma Function for Non-Negative Integer Argument | https://proofwiki.org/wiki/Gamma_Function_for_Non-Negative_Integer_Argument | https://proofwiki.org/wiki/Gamma_Function_for_Non-Negative_Integer_Argument | [
"Gamma Function"
] | [
"Definition:Gamma Function",
"Definition:Negative/Integer"
] | [
"Gamma Difference Equation"
] |
proofwiki-9425 | Gamma Function of Positive Half-Integer | {{begin-eqn}}
{{eqn | l = \map \Gamma {m + \frac 1 2}
| r = \frac {\paren {2 m}!} {2^{2 m} m!} \sqrt \pi
| c =
}}
{{eqn | r = \frac {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } {2^m} \sqrt \pi
| c =
}}
{{end-eqn}}
where:
:$m + \dfrac 1 2$ is a half-integer such that $m > 0$
:$\Gamma$... | Proof by induction:
For all $m \in \Z_{> 0}$, let $\map P m$ be the proposition:
:$\map \Gamma {m + \dfrac 1 2} = \dfrac {\paren {2 m}!} {2^{2 m} m!} \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {m + \frac 1 2}
| r = \frac {\paren {2 m}!} {2^{2 m} m!} \sqrt \pi
| c =
}}
{{eqn | r = \frac {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } {2^m} \sqrt \pi
| c =
}}
{{end-eqn}}
where:
:$m + \dfrac 1 2$ is a [[Definition:Half-Integer|half-integer]]... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $m \in \Z_{> 0}$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\map \Gamma {m + \dfrac 1 2} = \dfrac {\paren {2 m}!} {2^{2 m} m!} \sqrt \pi$ | Gamma Function of Positive Half-Integer | https://proofwiki.org/wiki/Gamma_Function_of_Positive_Half-Integer | https://proofwiki.org/wiki/Gamma_Function_of_Positive_Half-Integer | [
"Gamma Function"
] | [
"Definition:Half-Integer",
"Definition:Gamma Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-9426 | Gamma Function of Negative Half-Integer | {{begin-eqn}}
{{eqn | l = \map \Gamma {-m + \frac 1 2}
| r = \frac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi
| c =
}}
{{eqn | r = \frac {\paren {-1}^m 2^m} {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } \sqrt \pi
| c =
}}
{{end-eqn}}
where:
:$-m + \dfrac 1 2$ is a half-integ... | Proof by induction:
For all $m \in \Z_{> 0}$, let $\map P m$ be the proposition:
:$\map \Gamma {-m + \dfrac 1 2} = \dfrac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {-m + \frac 1 2}
| r = \frac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi
| c =
}}
{{eqn | r = \frac {\paren {-1}^m 2^m} {1 \times 3 \times 5 \times \cdots \times \paren {2 m - 1} } \sqrt \pi
| c =
}}
{{end-eqn}}
where:
:$-m + \dfrac 1 2$ is a [[Definiti... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $m \in \Z_{> 0}$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\map \Gamma {-m + \dfrac 1 2} = \dfrac {\paren {-1}^m 2^{2 m} m!} {\paren {2 m}!} \sqrt \pi$ | Gamma Function of Negative Half-Integer | https://proofwiki.org/wiki/Gamma_Function_of_Negative_Half-Integer | https://proofwiki.org/wiki/Gamma_Function_of_Negative_Half-Integer | [
"Gamma Function"
] | [
"Definition:Half-Integer",
"Definition:Gamma Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-9427 | Reciprocal times Derivative of Gamma Function | Let $z \in \C \setminus \Z_{\le 0}$.
Then:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
where:
:$\map \Gamma z$ denotes the Gamma function
:$\map {\Gamma'} z$ denotes the derivative of the Gamma function
:$\gamma$ denotes the Euler-Ma... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = Weierstrass Form of Gamma Function
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-\gamma z} } z \prod_{n \mathop = 1}^\infty \frac {... | Let $z \in \C \setminus \Z_{\le 0}$.
Then:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
where:
:$\map \Gamma z$ denotes the [[Definition:Gamma Function|Gamma function]]
:$\map {\Gamma'} z$ denotes the [[Definition:Derivative|deriva... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = [[Definition:Weierstrass Form of Gamma Function|Weierstrass Form of Gamma Function]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-... | Reciprocal times Derivative of Gamma Function/Proof 1 | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function/Proof_1 | [
"Reciprocal times Derivative of Gamma Function",
"Digamma Function",
"Gamma Function",
"General Harmonic Numbers"
] | [
"Definition:Gamma Function",
"Definition:Derivative",
"Definition:Gamma Function",
"Definition:Euler-Mascheroni Constant"
] | [
"Definition:Gamma Function/Weierstrass Form",
"Definition:Reciprocal",
"Definition:Differentiation",
"Product Rule for Derivatives",
"Definition:Continued Product",
"Definition:Division/Field/Complex Numbers"
] |
proofwiki-9428 | Reciprocal times Derivative of Gamma Function | Let $z \in \C \setminus \Z_{\le 0}$.
Then:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
where:
:$\map \Gamma z$ denotes the Gamma function
:$\map {\Gamma'} z$ denotes the derivative of the Gamma function
:$\gamma$ denotes the Euler-Ma... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = Weierstrass Form of Gamma Function
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-\gamma z} } z \prod_{n \mathop = 1}^\infty \frac {... | Let $z \in \C \setminus \Z_{\le 0}$.
Then:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
where:
:$\map \Gamma z$ denotes the [[Definition:Gamma Function|Gamma function]]
:$\map {\Gamma'} z$ denotes the [[Definition:Derivative|deriva... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = [[Definition:Weierstrass Form of Gamma Function|Weierstrass Form of Gamma Function]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-... | Reciprocal times Derivative of Gamma Function/Proof 2 | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function/Proof_2 | [
"Reciprocal times Derivative of Gamma Function",
"Digamma Function",
"Gamma Function",
"General Harmonic Numbers"
] | [
"Definition:Gamma Function",
"Definition:Derivative",
"Definition:Gamma Function",
"Definition:Euler-Mascheroni Constant"
] | [
"Definition:Gamma Function/Weierstrass Form",
"Definition:Reciprocal",
"Definition:Natural Logarithm",
"Sum of Logarithms",
"Difference of Logarithms",
"Logarithm of Power",
"Natural Logarithm of e is 1",
"Definition:Differentiation",
"Derivative of Composite Function",
"Derivative of Natural Loga... |
proofwiki-9429 | Derivative of Gamma Function at 1 | Let $\Gamma$ denote the Gamma function.
Then:
:$\map {\Gamma'} 1 = \ds \int_0^\infty e^{-x} \ln x \rd x = -\gamma$
where:
:$\map {\Gamma'} 1$ denotes the derivative of the Gamma function evaluated at $1$
:$\gamma$ denotes the Euler-Mascheroni constant. | From Reciprocal times Derivative of Gamma Function:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
Setting $z = 1$:
{{begin-eqn}}
{{eqn | l = \frac {\map {\Gamma'} 1} {\map \Gamma 1}
| r = -\gamma + \sum_{n \mathop = 1}^\infty \pa... | Let $\Gamma$ denote the [[Definition:Gamma Function|Gamma function]].
Then:
:$\map {\Gamma'} 1 = \ds \int_0^\infty e^{-x} \ln x \rd x = -\gamma$
where:
:$\map {\Gamma'} 1$ denotes the [[Definition:Derivative|derivative]] of the [[Definition:Gamma Function|Gamma function]] evaluated at $1$
:$\gamma$ denotes the [[De... | From [[Reciprocal times Derivative of Gamma Function]]:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
Setting $z = 1$:
{{begin-eqn}}
{{eqn | l = \frac {\map {\Gamma'} 1} {\map \Gamma 1}
| r = -\gamma + \sum_{n \mathop = 1}^\in... | Derivative of Gamma Function at 1/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Gamma_Function_at_1 | https://proofwiki.org/wiki/Derivative_of_Gamma_Function_at_1/Proof_1 | [
"Derivative of Gamma Function at 1",
"Gamma Function",
"Euler-Mascheroni Constant"
] | [
"Definition:Gamma Function",
"Definition:Derivative",
"Definition:Gamma Function",
"Definition:Euler-Mascheroni Constant"
] | [
"Reciprocal times Derivative of Gamma Function",
"Gamma Function Extends Factorial"
] |
proofwiki-9430 | Stirling's Formula for Gamma Function | Let $\Gamma$ denote the Gamma function.
Let $z \in \C$ with a strictly positive real part and $\size {\arg z} < \dfrac \pi 2$.
Then:
:$\map \Gamma {z + 1} = \sqrt {2 \pi z} \, z^z e^{-z} \paren {1 + \dfrac 1 {12 z} + \dfrac 1 {288 z^2} - \dfrac {139} {51 \, 840 z^3} - \dfrac {571} {2\, 488 \, 320 z^4} + \dfrac {163 \, ... | From Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function, we have:
:$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {\dfrac 1 {z^{2 d + 1} } }$
Taking the... | Let $\Gamma$ denote the [[Definition:Gamma Function|Gamma function]].
Let $z \in \C$ with a [[Definition:Strictly Positive Real Number|strictly positive]] [[Definition:Real Part|real part]] and $\size {\arg z} < \dfrac \pi 2$.
Then:
:$\map \Gamma {z + 1} = \sqrt {2 \pi z} \, z^z e^{-z} \paren {1 + \dfrac 1 {12 z} + ... | From [[Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function]], we have:
:$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {\dfrac 1 {z^{2 d + 1} } }$
Takin... | Stirling's Formula for Gamma Function | https://proofwiki.org/wiki/Stirling's_Formula_for_Gamma_Function | https://proofwiki.org/wiki/Stirling's_Formula_for_Gamma_Function | [
"Stirling's Formula",
"Asymptotic Expansions",
"Gamma Function"
] | [
"Definition:Gamma Function",
"Definition:Strictly Positive/Real Number",
"Definition:Complex Number/Real Part"
] | [
"Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function",
"Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function",
"Definition:Exponential Function/Real/Inverse of Natural Logarithm",
"Logarithm of Power/Natural Logarithm",
"Definition:Exponential Functio... |
proofwiki-9431 | Commutativity of Parameters of Beta Function | :$\map \Beta {x, y} = \map \Beta {y, x}$ | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \frac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }
| c = {{Defof|Beta Function|index = 3}}
}}
{{eqn | r = \frac {\map \Gamma y \map \Gamma x} {\map \Gamma {y + x} }
| c = Commutative Law of Addition and Commutative Law of Multiplication
}}
{{eq... | :$\map \Beta {x, y} = \map \Beta {y, x}$ | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \frac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }
| c = {{Defof|Beta Function|index = 3}}
}}
{{eqn | r = \frac {\map \Gamma y \map \Gamma x} {\map \Gamma {y + x} }
| c = [[Commutative Law of Addition]] and [[Commutative Law of Multiplication]]... | Commutativity of Parameters of Beta Function | https://proofwiki.org/wiki/Commutativity_of_Parameters_of_Beta_Function | https://proofwiki.org/wiki/Commutativity_of_Parameters_of_Beta_Function | [
"Beta Function"
] | [] | [
"Commutative Law of Addition",
"Commutative Law of Multiplication"
] |
proofwiki-9432 | Beta Function as Integral of Power of t over Power of t plus 1 | :$\ds \map \Beta {x, y} = \int_{\mathop \to 0}^{\mathop \to \infty} \frac {t^{x - 1} } {\paren {1 + t}^{x + y} } \rd t$
where $\Beta$ denotes the Beta function. | Consider the substitution $s = \dfrac t {1 + t}$.
We have the following:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + t}
| r = 1 - s
}}
{{eqn | l = t \to 0
| o = \implies
| r = s \to 0
}}
{{eqn | l = t \to \infty
| o = \implies
| r = s \to 1
}}
{{eqn | l = \d s
| r = \dfrac 1 {\paren {1 +... | :$\ds \map \Beta {x, y} = \int_{\mathop \to 0}^{\mathop \to \infty} \frac {t^{x - 1} } {\paren {1 + t}^{x + y} } \rd t$
where $\Beta$ denotes the [[Definition:Beta Function|Beta function]]. | Consider the [[Integration by Substitution|substitution]] $s = \dfrac t {1 + t}$.
We have the following:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + t}
| r = 1 - s
}}
{{eqn | l = t \to 0
| o = \implies
| r = s \to 0
}}
{{eqn | l = t \to \infty
| o = \implies
| r = s \to 1
}}
{{eqn | l = \d s... | Beta Function as Integral of Power of t over Power of t plus 1 | https://proofwiki.org/wiki/Beta_Function_as_Integral_of_Power_of_t_over_Power_of_t_plus_1 | https://proofwiki.org/wiki/Beta_Function_as_Integral_of_Power_of_t_over_Power_of_t_plus_1 | [
"Beta Function as Integral of Power of t over Power of t plus 1",
"Beta Function"
] | [
"Definition:Beta Function"
] | [
"Integration by Substitution",
"Integration by Substitution"
] |
proofwiki-9433 | Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t | :$\ds \map \Beta {x, y} := r^y \paren {r + 1}^x \int_{\mathop \to 0}^{\mathop \to 1} \frac {t^{x - 1} \paren {1 - t}^{y - 1} } {\paren {r + t}^{x + y} } \rd t$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\paren {r + 1} t} {\paren {r + t} }
| c = Integration by Substitution
}}
{{eqn | ll= \leadsto
| l = \rd u
| r = \frac {\paren {r + t} \paren {r + 1} - \paren {r + 1} t} {\paren {r + t}^2} \rd t
| c = Quotient Rule for Derivatives
}}
{{eqn | r =... | :$\ds \map \Beta {x, y} := r^y \paren {r + 1}^x \int_{\mathop \to 0}^{\mathop \to 1} \frac {t^{x - 1} \paren {1 - t}^{y - 1} } {\paren {r + t}^{x + y} } \rd t$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \frac {\paren {r + 1} t} {\paren {r + t} }
| c = [[Integration by Substitution]]
}}
{{eqn | ll= \leadsto
| l = \rd u
| r = \frac {\paren {r + t} \paren {r + 1} - \paren {r + 1} t} {\paren {r + t}^2} \rd t
| c = [[Quotient Rule for Derivatives]]
}}
{{e... | Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t | https://proofwiki.org/wiki/Beta_Function_as_Integral_of_Power_of_t_by_Power_of_1_minus_t_over_Power_of_r_plus_t | https://proofwiki.org/wiki/Beta_Function_as_Integral_of_Power_of_t_by_Power_of_1_minus_t_over_Power_of_r_plus_t | [
"Beta Function"
] | [] | [
"Integration by Substitution",
"Quotient Rule for Derivatives",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Power of Product",
"Integration by Substitution"
] |
proofwiki-9434 | Solution to Separable Differential Equation/General Result | Consider the separable differential equation:
:$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$
Its general solution is found by solving the integration:
:$\ds \int \frac {\map {g_1} x} {\map {g_2} x} \rd x + \int \frac {\map {h_2} y} {\map {h_1} y} \rd y = C$ | {{begin-eqn}}
{{eqn | l = \map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \frac {\d y} {\d x}
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\map {g_1} x} {\map {g_2} x} + \frac {\map {h_2} y} {\map {h_1} y} \frac {\d y} {\d x}
| r = 0
| c = dividing both sides by $\map {g_2} x ... | Consider the [[Definition:Separable Differential Equation|separable differential equation]]:
:$\map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \dfrac {\d y} {\d x} = 0$
Its [[Definition:General Solution to Differential Equation|general solution]] is found by solving the [[Definition:Integration|integration]]:
:... | {{begin-eqn}}
{{eqn | l = \map {g_1} x \map {h_1} y + \map {g_2} x \map {h_2} y \frac {\d y} {\d x}
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\map {g_1} x} {\map {g_2} x} + \frac {\map {h_2} y} {\map {h_1} y} \frac {\d y} {\d x}
| r = 0
| c = dividing both sides by $\map {g_2} x ... | Solution to Separable Differential Equation/General Result | https://proofwiki.org/wiki/Solution_to_Separable_Differential_Equation/General_Result | https://proofwiki.org/wiki/Solution_to_Separable_Differential_Equation/General_Result | [
"Solution to Separable Differential Equation"
] | [
"Definition:Separable Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Primitive (Calculus)/Integration"
] | [
"Definition:Primitive (Calculus)/Integration",
"Primitive of Constant",
"Linear Combination of Integrals/Indefinite",
"Integration by Substitution"
] |
proofwiki-9435 | Sum of Arithmetic-Geometric Sequence | Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as:
:$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$
Then its closed-form expression is:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n... | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-... | Let $\sequence {a_k}$ be an [[Definition:Arithmetic-Geometric Sequence|arithmetic-geometric sequence]] defined as:
:$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$
Then its [[Definition:Closed-Form Expression|closed-form expression]] is:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^... | Sum of Arithmetic-Geometric Sequence/Proof 1 | https://proofwiki.org/wiki/Sum_of_Arithmetic-Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Arithmetic-Geometric_Sequence/Proof_1 | [
"Sums of Sequences",
"Sum of Arithmetic-Geometric Sequence",
"Arithmetic-Geometric Sequences"
] | [
"Definition:Arithmetic-Geometric Sequence",
"Definition:Closed Form Expression"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Sum of Arithmetic-Geometric Sequence",
"Definition:Fraction/Denominator",
"Principle of Mathematical Induction"
] |
proofwiki-9436 | Sum of Arithmetic-Geometric Sequence | Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as:
:$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$
Then its closed-form expression is:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k
| r = a \sum_{k \mathop = 0}^{n - 1} r^k + d \sum_{k \mathop = 0}^{n - 1} k r^k
| c =
}}
{{eqn | r = \frac {a \paren {1 - r^n} } {1 - r} + d \sum_{k \mathop = 0}^{n - 1} k r^k
| c = Sum of Geometric Sequence
}}
{{eqn | r = \f... | Let $\sequence {a_k}$ be an [[Definition:Arithmetic-Geometric Sequence|arithmetic-geometric sequence]] defined as:
:$a_k = \paren {a + k d} r^k$ for $k = 0, 1, 2, \ldots, n - 1$
Then its [[Definition:Closed-Form Expression|closed-form expression]] is:
:$\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k
| r = a \sum_{k \mathop = 0}^{n - 1} r^k + d \sum_{k \mathop = 0}^{n - 1} k r^k
| c =
}}
{{eqn | r = \frac {a \paren {1 - r^n} } {1 - r} + d \sum_{k \mathop = 0}^{n - 1} k r^k
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | r ... | Sum of Arithmetic-Geometric Sequence/Proof 2 | https://proofwiki.org/wiki/Sum_of_Arithmetic-Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Arithmetic-Geometric_Sequence/Proof_2 | [
"Sums of Sequences",
"Sum of Arithmetic-Geometric Sequence",
"Arithmetic-Geometric Sequences"
] | [
"Definition:Arithmetic-Geometric Sequence",
"Definition:Closed Form Expression"
] | [
"Sum of Geometric Sequence",
"Sum of Sequence of Power by Index"
] |
proofwiki-9437 | Sum of Infinite Arithmetic-Geometric Sequence | Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as:
:$a_k = \paren {a + k d} r^k$ for $n = 0, 1, 2, \ldots$
Let:
:$\size r < 1$
where $\size r$ denotes the absolute value of $r$.
Then:
:$\ds \sum_{n \mathop = 0}^\infty \paren {a + k d} r^k = \frac a {1 - r} + \frac {r d} {\paren {1 - r}^2}$ | From Sum of Arithmetic-Geometric Sequence, we have:
:$\ds s_n = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$
We have that $\size r < 1$.
So by Sequence of Powers of Number less than One:
:$r^n \to... | Let $\sequence {a_k}$ be an [[Definition:Arithmetic-Geometric Sequence|arithmetic-geometric sequence]] defined as:
:$a_k = \paren {a + k d} r^k$ for $n = 0, 1, 2, \ldots$
Let:
:$\size r < 1$
where $\size r$ denotes the [[Definition:Absolute Value|absolute value]] of $r$.
Then:
:$\ds \sum_{n \mathop = 0}^\infty \pare... | From [[Sum of Arithmetic-Geometric Sequence]], we have:
:$\ds s_n = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k = \frac {a \paren {1 - r^n} } {1 - r} + \frac {r d \paren {1 - n r^{n - 1} + \paren {n - 1} r^n} } {\paren {1 - r}^2}$
We have that $\size r < 1$.
So by [[Sequence of Powers of Number less than One]]:... | Sum of Infinite Arithmetic-Geometric Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Arithmetic-Geometric_Sequence | https://proofwiki.org/wiki/Sum_of_Infinite_Arithmetic-Geometric_Sequence | [
"Sums of Sequences",
"Arithmetic-Geometric Sequences"
] | [
"Definition:Arithmetic-Geometric Sequence",
"Definition:Absolute Value"
] | [
"Sum of Arithmetic-Geometric Sequence",
"Sequence of Powers of Number less than One"
] |
proofwiki-9438 | Reduction Formula for Integral of Power of Cosine | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let:
:$I_n := \ds \int \cos^n x \rd x$
Then:
:$I_n = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n - 2}$
is a reduction formula for $\ds \int \cos^n x \rd x$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\rd u} {\rd x}
| r = -\paren {n - 1} \cos ^{n - 2} x \sin x
... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let:
:$I_n := \ds \int \cos^n x \rd x$
Then:
:$I_n = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n - 2}$
is a [[Definition:Reduction Formula (Calculus)|reduction formula]] for $\ds \int \cos^n x \rd x$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\rd v} {\rd x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \cos^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\rd u} {\rd x}
| r = -\paren {n - 1} \cos ^{n - 2} x \sin x
... | Reduction Formula for Integral of Power of Cosine | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Cosine | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Cosine | [
"Reduction Formula for Integral of Power of Cosine",
"Reduction Formulae (Calculus)",
"Primitives involving Cosine Function"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Reduction Formula (Calculus)"
] | [
"Derivative of Composite Function",
"Derivative of Cosine Function",
"Power Rule for Derivatives",
"Primitive of Cosine Function",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9439 | Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0 | :$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {2 \sqrt {b x + c} } b + C$
when $a = 0$. | {{begin-eqn}}
{{eqn | l = a
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sqrt {a x^2 + b x + c} }
| r = \int \frac {\d x} {\sqrt {b x + c} }
| c =
}}
{{eqn | r = \frac {2 \sqrt {b x + c} } b
| c = Primitive of $\dfrac 1 {\sqrt {a x + b} }$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {\sqrt {a x^2 + b x + c} } = \frac {2 \sqrt {b x + c} } b + C$
when $a = 0$. | {{begin-eqn}}
{{eqn | l = a
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\sqrt {a x^2 + b x + c} }
| r = \int \frac {\d x} {\sqrt {b x + c} }
| c =
}}
{{eqn | r = \frac {2 \sqrt {b x + c} } b
| c = [[Primitive of Reciprocal of Root of a x + b|Primitive of $\dfrac ... | Primitive of Reciprocal of Root of a x squared plus b x plus c/a equal to 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_equal_to_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/a_equal_to_0 | [
"Primitive of Reciprocal of Root of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of Root of a x + b"
] |
proofwiki-9440 | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1 | Let $a \in \R_{\ne 0}$.
{{:Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0}} | First:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\d x} {x^2 + \frac c a}
| c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
Let $a c > 0$.
Then $\dfrac c a > 0$ and:
{{begin-eqn}}
{{eqn | l... | Let $a \in \R_{\ne 0}$.
{{:Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0}} | First:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\d x} {x^2 + \frac c a}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{end-eqn}}
Let $a c > 0$.
Then $\dfrac c a > 0$ and:
{{begin-eqn}}
{... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_1 | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Constant Multiple of Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form",
"Primitive of Power"
] |
proofwiki-9441 | Primitive of x over Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } = \frac {\sqrt {a x^2 + b x + c} } a - \frac b {2 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = Derivative of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} }
| r = \frac 1 {2 a} \int \frac {2... | Let $a \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$:
:$\ds \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} } = \frac {\sqrt {a x^2 + b x + c} } a - \frac b {2 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = [[Derivative of Power]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sqrt {a x^2 + b x + c} }
| r = \frac 1 {2 a} \int \f... | Primitive of x over Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitive of x over Root of a x squared plus b x plus c",
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Linear Combination of Integrals/Indefinite",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9442 | Primitive of x squared over Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} } = \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = Derivative of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} }
| c =
}}
{{eqn | ... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} } = \frac {2 a x - 3 b} {4 a^2} \sqrt {a x^2 + b x + c} + \frac {3 b^2 - 4 a c} {8 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = [[Derivative of Power]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\sqrt {a x^2 + b x + c} }
| c =
}}
{{... | Primitive of x squared over Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Linear Combination of Integrals/Indefinite",
"Primitive of Root of a x squared plus b x plus c",
"Primitive of x over Root of a x squared plus b x plus c",
"Definition:Common Denominator"
] |
proofwiki-9443 | Primitive of Reciprocal of x by Root of a x squared plus b x plus c | Let $a, b, c \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $x \ne 0$:
$\quad \ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac {-1} {\sqrt c} \dfrac {\size x} x \ln \size {\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x} + C & : c > 0, b^2 - 4 a c > 0 \\
\dfr... | === Lemma ===
{{:Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma}}{{qed|lemma}}
Let $x > 0$, and so $u > 0$.
Then we have:
:$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\sqrt {a + b u + c u^2} }$
We consider the two cases where $c > 0$ and $c < 0$.
First we take $c... | Let $a, b, c \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $x \ne 0$:
$\quad \ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac {-1} {\sqrt c} \dfrac {\size x} x \ln \size {\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x} + C & : c > 0, b^2 - 4 a c > 0 \\
\d... | === [[Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma}}{{qed|lemma}}
Let $x > 0$, and so $u > 0$.
Then we have:
:$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\sqrt {a + b u... | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Proof_1 | [
"Primitives involving Root of a x squared plus b x plus c",
"Primitive of Reciprocal of x by Root of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a grea... |
proofwiki-9444 | Primitive of Reciprocal of x by Root of a x squared plus b x plus c | Let $a, b, c \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $x \ne 0$:
$\quad \ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac {-1} {\sqrt c} \dfrac {\size x} x \ln \size {\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x} + C & : c > 0, b^2 - 4 a c > 0 \\
\dfr... | {{begin-eqn}}
{{eqn | l = x \sqrt {a x^2 + b x + c}
| r = \frac x {\paren {a x^2 + b x + c}^{-\frac 1 2} }
| c =
}}
{{eqn | r = \frac{x \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} } {\paren {a x^2 + b x + c}^{-\frac 1 2} \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} }
| c =
}}
{{... | Let $a, b, c \in \R_{\ne 0}$.
Then for $x \in \R$ such that $a x^2 + b x + c > 0$ and $x \ne 0$:
$\quad \ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = \begin {cases}
\dfrac {-1} {\sqrt c} \dfrac {\size x} x \ln \size {\dfrac {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} x} + C & : c > 0, b^2 - 4 a c > 0 \\
\d... | {{begin-eqn}}
{{eqn | l = x \sqrt {a x^2 + b x + c}
| r = \frac x {\paren {a x^2 + b x + c}^{-\frac 1 2} }
| c =
}}
{{eqn | r = \frac{x \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} } {\paren {a x^2 + b x + c}^{-\frac 1 2} \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} }
| c =
}}
{{... | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Proof_2 | [
"Primitives involving Root of a x squared plus b x plus c",
"Primitive of Reciprocal of x by Root of a x squared plus b x plus c"
] | [] | [
"Definition:Real Number",
"Primitive of Function under its Derivative"
] |
proofwiki-9445 | Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} } = -\frac {\sqrt {a x^2 + b x + c} } {c x} - \frac b {2 c} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} }
}}
{{eqn | r = \int \frac {c \d x} {c x^2 \sqrt {a x^2 + b x + c} }
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} \rd x} {c x^2 \sqrt {a x^2 + b x + c} }
| c = ... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} } = -\frac {\sqrt {a x^2 + b x + c} } {c x} - \frac b {2 c} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \sqrt {a x^2 + b x + c} }
}}
{{eqn | r = \int \frac {c \d x} {c x^2 \sqrt {a x^2 + b x + c} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} ... | Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Root of a x squared plus b x plus c over x squared"
] |
proofwiki-9446 | Primitive of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = \paren {2 a x + b}^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 4 a \paren {2 a x + b}
| c = Derivative of Power and Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 4 a \sqrt z
|... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = \paren {2 a x + b}^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 4 a \paren {2 a x + b}
| c = [[Derivative of Power]] and [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 4 a \sqrt z... | Primitive of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Completing the Square",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of p x + q over Root of a x + b",
"Definition:Square Root",
"Definition:Negative/Number",
"Completing the Square"... |
proofwiki-9447 | Primitive of x by Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac {b \paren {2 a x + b} \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \paren {4 a c - b^2} } {16 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | o =
| r = \int x \sqrt {a x^2 + b x + c} \rd x
}}
{{eqn | r = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x
| c = Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$
}}
{{eqn | r = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a}
... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac {b \paren {2 a x + b} \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \paren {4 a c - b^2} } {16 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | o =
| r = \int x \sqrt {a x^2 + b x + c} \rd x
}}
{{eqn | r = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x
| c = [[Primitive of x by Root of a x squared plus b x plus c/Lemma|Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$]]
}... | Primitive of x by Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Primitive of x by Root of a x squared plus b x plus c/Lemma",
"Primitive of Root of a x squared plus b x plus c"
] |
proofwiki-9448 | Primitive of x squared by Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ | {{begin-eqn}}
{{eqn | o =
| r = \int x^2 \sqrt {a x^2 + b x + c} \rd x
}}
{{eqn | r = \int \frac {2 a x^2} {2 a} \sqrt {a x^2 + b x + c} \rd x
| c = multiplying top and bottom by $2 a$
}}
{{eqn | r = \int \frac {x \paren {2 a x + b - b} } {2 a} \sqrt {a x^2 + b x + c} \rd x
}}
{{eqn | n = 1
| r = \fr... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$ | {{begin-eqn}}
{{eqn | o =
| r = \int x^2 \sqrt {a x^2 + b x + c} \rd x
}}
{{eqn | r = \int \frac {2 a x^2} {2 a} \sqrt {a x^2 + b x + c} \rd x
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $2 a$
}}
{{eqn | r = \int \frac {x \paren {2 a x + b - b} } {2 a} \sqrt {a x... | Primitive of x squared by Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power",
"Definition:Primitive (Calculus)",
"Derivative of Identity Function",
"Integration by Parts",
"Linear... |
proofwiki-9449 | Primitive of Root of a x squared plus b x plus c over x | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\sqrt {a x^2 + b x + c} } x \rd x = \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\sqrt {a x^2 + b x + c} } x \rd x
| o =
}}
{{eqn | r = \int \frac {a x^2 + b x + c} {x \sqrt {a x^2 + b x + c} } \rd x
| c = multiplying top and bottom by $\sqrt {a x^2 + b x + c}$
}}
{{eqn | r = a \int \frac {x^2 \rd x} {x \sqrt {a x^2 + b x + c} } + b \int \frac {x \... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\sqrt {a x^2 + b x + c} } x \rd x = \sqrt {a x^2 + b x + c} + \frac b 2 \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\sqrt {a x^2 + b x + c} } x \rd x
| o =
}}
{{eqn | r = \int \frac {a x^2 + b x + c} {x \sqrt {a x^2 + b x + c} } \rd x
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\sqrt {a x^2 + b x + c}$
}}
{{eqn | r = a \int \frac {x^2 \rd... | Primitive of Root of a x squared plus b x plus c over x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c_over_x | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Root of a x squared plus b x plus c"
] |
proofwiki-9450 | Primitive of Root of a x squared plus b x plus c over x squared | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \rd x = \frac {-\sqrt {a x^2 + b x + c} } x + a \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + \frac b 2 \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \rd x
| o =
}}
{{eqn | r = \int \frac {a x^2 + b x + c} {x^2 \sqrt {a x^2 + b x + c} } \rd x
| c = multiplying top and bottom by $\sqrt {a x^2 + b x + c}$
}}
{{eqn | r = a \int \frac {x^2 \rd x} {x^2 \sqrt {a x^2 + b x + c} } + b \int \f... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \rd x = \frac {-\sqrt {a x^2 + b x + c} } x + a \int \frac {\d x} {\sqrt {a x^2 + b x + c} } + \frac b 2 \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\sqrt {a x^2 + b x + c} } {x^2} \rd x
| o =
}}
{{eqn | r = \int \frac {a x^2 + b x + c} {x^2 \sqrt {a x^2 + b x + c} } \rd x
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\sqrt {a x^2 + b x + c}$
}}
{{eqn | r = a \int \frac {x... | Primitive of Root of a x squared plus b x plus c over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_squared_plus_b_x_plus_c_over_x_squared | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c"
] |
proofwiki-9451 | Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {2 a x + b} } {\paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + C$ | For $a > 0$:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| r = \int \frac {\d x} {\paren {\sqrt {\frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a} } }^3}
| c = Completing the Square
}}
{{eqn | r = \int \frac {8 a \sqrt a \rd x} {\paren {\sqrt {\paren {2 a x + b}^2 + 4 a ... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {2 a x + b} } {\paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + C$ | For $a > 0$:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| r = \int \frac {\d x} {\paren {\sqrt {\frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a} } }^3}
| c = [[Completing the Square]]
}}
{{eqn | r = \int \frac {8 a \sqrt a \rd x} {\paren {\sqrt {\paren {2 a x + b}^2 +... | Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Completing the Square",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of x squared plus a squared cubed",
"Primitive of Reciprocal of Root of x squared minus a squared cubed",
"Completing the Square"
] |
proofwiki-9452 | Primitive of x over Cube of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {b x + 2 c} } {\paren {b^2 - 4 a c} \sqrt {a x^2 + b x + c} } + C$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = Derivative of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| r = \frac 1 {2 a} \i... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {b x + 2 c} } {\paren {b^2 - 4 a c} \sqrt {a x^2 + b x + c} } + C$ | First:
{{begin-eqn}}
{{eqn | l = z
| r = a x^2 + b x + c
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 a x + b
| c = [[Derivative of Power]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| r = \frac 1 {2... | Primitive of x over Cube of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_over_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_over_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Power Rule for Derivatives",
"Linear Combination of Integrals/Indefinite",
"Integration by Substitution",
"Primitive of Power",
"Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c"
] |
proofwiki-9453 | Primitive of x squared over Cube of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {\paren {2 b^2 - 4 a c} x + 2 b c} {a \paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {a x^2 \rd x} {a \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying top and bottom by $a$
}}
{{eqn | r = \int \frac {\paren {a x^2 + b x + c - b x - c} \rd x} {a \paren {\... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {\paren {2 b^2 - 4 a c} x + 2 b c} {a \paren {4 a c - b^2} \sqrt {a x^2 + b x + c} } + \frac 1 a \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {x^2 \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {a x^2 \rd x} {a \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a$
}}
{{eqn | r = \int \frac {\p... | Primitive of x squared over Cube of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Cube of Root of a x squared plus b x plus c",
"Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c"
] |
proofwiki-9454 | Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {\sqrt {a x^2 + b x + c} }^3} = \frac 1 {c \sqrt {a x^2 + b x + c} } + \frac 1 c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x \paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} \rd x} {c x \paren {... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {\sqrt {a x^2 + b x + c} }^3} = \frac 1 {c \sqrt {a x^2 + b x + c} } + \frac 1 c \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } - \frac b {2 c} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3}$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x \paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \int \frac {\paren ... | Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Cube of Root of a x squared plus b x plus c",
"Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c",
"Definition:Common Denominator",
"Primitive of Re... |
proofwiki-9455 | Primitive of Reciprocal of x squared by Cube of Root of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} = -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \int \frac {\paren {a x^2 + b x + c - a x^2 - b x} \rd x} {c x^2 \p... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3} = -\frac {a x^2 + 2 b x + c} {c^2 x \sqrt {a x^2 + b x + c} } + \frac {b^2 - 2 a c} {2 c^2} \int \frac {\d x} {\paren {\sqrt {a x^2 + b x + c} }^3} - \frac {3 b} {2 c^2} \int \frac {\d x} {x \sqrt {a x^2 + b x + c} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x^2 \paren {\sqrt {a x^2 + b x + c} }^3}
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {\sqrt {a x^2 + b x + c} }^3}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \int \frac {\pa... | Primitive of Reciprocal of x squared by Cube of Root of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Cube_of_Root_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by Root of a x squared plus b x plus c",
"Primitive of Reciprocal of x by Cube of Root of a x squared plus b x plus c"
] |
proofwiki-9456 | Primitive of Half Integer Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} } {4 a \paren {n + 1} } + \frac {\paren {2 n + 1} \paren {4 a c - b^2} } {8 a \paren {n + 1} } \int \paren {a x^2 + b x + c}^{n - \frac 1 2} \rd x$ | {{finish|This only takes on the case where $a > 0$. The case where $a < 0$ needs to be addressed.}}
{{begin-eqn}}
{{eqn | l = \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x
| r = \int \paren {\frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a} }^{n + \frac 1 2} \rd x
| c = Completing the Square
}}
{{eqn |... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} } {4 a \paren {n + 1} } + \frac {\paren {2 n + 1} \paren {4 a c - b^2} } {8 a \paren {n + 1} } \int \paren {a x^2 + b x + c}^{n - \frac 1 2} \rd x$ | {{finish|This only takes on the case where $a > 0$. The case where $a < 0$ needs to be addressed.}}
{{begin-eqn}}
{{eqn | l = \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x
| r = \int \paren {\frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a} }^{n + \frac 1 2} \rd x
| c = [[Completing the Square]]
}}
{{... | Primitive of Half Integer Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Completing the Square",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power of p x + q over Root of a x + b",
"Primitive of Power of p x + q ove... |
proofwiki-9457 | Primitive of x by Half Integer Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x$ | {{begin-eqn}}
{{eqn | o =
| r = \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x
}}
{{eqn | r = \int \frac {2 a x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x} {2 a}
| c = multiplying top and bottom by $2 a$
}}
{{eqn | r = \int \frac {\paren {2 a x + b - b} \paren {a x^2 + b x + c}^{n + \frac 1 2} \... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x$ | {{begin-eqn}}
{{eqn | o =
| r = \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x
}}
{{eqn | r = \int \frac {2 a x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x} {2 a}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $2 a$
}}
{{eqn | r = \int \frac {\paren {2 a x... | Primitive of x by Half Integer Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_by_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_by_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9458 | Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac {2 \paren {2 a x + b} } {\paren {2 n - 1} \paren {4 a c - b^2} \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac {8 a \paren {n - 1} } {\paren {2 n - 1} \paren {4 a c - b^2} } \int \frac {\d x} {\paren {a x^2 + b ... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }
| r = \int \paren {\frac {4 a} {\paren {2 a x + b}^2 + 4 a c - b^2} }^{n + \frac 1 2} \rd x
| c = Completing the Square
}}
{{eqn | r = \paren {2 \sqrt a}^{2 n + 1} \int \frac {\d x} {\paren {\paren {2 a x + b}^2 + 4 a c... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac {2 \paren {2 a x + b} } {\paren {2 n - 1} \paren {4 a c - b^2} \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac {8 a \paren {n - 1} } {\paren {2 n - 1} \paren {4 a c - b^2} } \int \frac {\d x} {\paren {a x^2 + b... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }
| r = \int \paren {\frac {4 a} {\paren {2 a x + b}^2 + 4 a c - b^2} }^{n + \frac 1 2} \rd x
| c = [[Completing the Square]]
}}
{{eqn | r = \paren {2 \sqrt a}^{2 n + 1} \int \frac {\d x} {\paren {\paren {2 a x + b}^2 + 4... | Primitive of Reciprocal of Half Integer Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Completing the Square",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of Power of p x + q by Root of a x + b",
"Primitive of Reciproc... |
proofwiki-9459 | Primitive of Reciprocal of x by Half Integer Power of a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac ... | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} }
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x \paren {a x^2 + b x + c}^{n + \frac 1 2} }
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \frac 1 c \int \frac {\paren {a x^2 + b x + c - a x^2 - b x... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac... | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} }
| o =
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x \paren {a x^2 + b x + c}^{n + \frac 1 2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \frac 1... | Primitive of Reciprocal of x by Half Integer Power of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Half_Integer_Power_of_a_x_squared_plus_b_x_plus_c | [
"Primitives involving Root of a x squared plus b x plus c"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive ... |
proofwiki-9460 | Primitive of x over x cubed plus a cubed | :$\ds \int \frac {x \rd x} {x^3 + a^3} = \frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {x^3 + a^3}
| r = \int \frac {\paren {x + a - a} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x + a} \rd x} {x^3 + a^3} - a \int \frac {\d x} {x^3 + a^3}
| c = Linear Combination of Primitives
}}
{{eqn | r = \int \frac {\paren {x + a} \rd ... | :$\ds \int \frac {x \rd x} {x^3 + a^3} = \frac 1 {6 a} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {a \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {x^3 + a^3}
| r = \int \frac {\paren {x + a - a} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x + a} \rd x} {x^3 + a^3} - a \int \frac {\d x} {x^3 + a^3}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = \int \frac {\paren {x + a} ... | Primitive of x over x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_x_over_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_x_cubed_plus_a_cubed | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes",
"Primitive of Reciprocal of x cubed plus a cubed/Lemma",
"Primitive of Reciprocal of x cubed plus a cubed"
] |
proofwiki-9461 | Primitive of Reciprocal of x cubed plus a cubed | :$\ds \int \frac {\d x} {x^3 + a^3} = \frac 1 {6 a^2} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 1 {a^2 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 + a^3}
| r = \int \paren {\frac 1 {3 a^2 \paren {x + a} } - \frac {x - 2 a} {3 a^2 \paren {x^2 - a x + a^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \int \paren {\frac 1 {3 a^2 \paren {x + a} } - \frac {2 x - 4 a} {6 a^2 \paren {x^2 - a x + a... | :$\ds \int \frac {\d x} {x^3 + a^3} = \frac 1 {6 a^2} \ln \size {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 1 {a^2 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 + a^3}
| r = \int \paren {\frac 1 {3 a^2 \paren {x + a} } - \frac {x - 2 a} {3 a^2 \paren {x^2 - a x + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x cubed plus a cubed/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \int \paren {\... | Primitive of Reciprocal of x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_plus_a_cubed | [
"Primitive of Reciprocal of x cubed plus a cubed",
"Primitives involving x cubed plus a cubed"
] | [] | [
"Primitive of Reciprocal of x cubed plus a cubed/Partial Fraction Expansion",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Primitive of Function under its Derivative",
"Logarithm of Power",
"Dif... |
proofwiki-9462 | Primitive of x squared over x cubed plus a cubed | :$\ds \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {x^3 + a^3}
| r = 3 x^2
| c = Derivative of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^2 \rd x} {x^3 + a^3}
| r = \frac 1 3 \ln \size {x^3 + a^3} + C
| c = Primitive of Function under its Derivative
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {x^3 + a^3}
| r = 3 x^2
| c = [[Derivative of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^2 \rd x} {x^3 + a^3}
| r = \frac 1 3 \ln \size {x^3 + a^3} + C
| c = [[Primitive of Function under its Derivative]]
}}
{{end-eqn}}
{{qed}} | Primitive of x squared over x cubed plus a cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed/Proof_1 | [
"Primitive of x squared over x cubed plus a cubed",
"Primitives involving x cubed plus a cubed"
] | [] | [
"Power Rule for Derivatives",
"Primitive of Function under its Derivative"
] |
proofwiki-9463 | Primitive of x squared over x cubed plus a cubed | :$\ds \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$ | From Primitive of Power of x less one over Power of x plus Power of a:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^3 + a^3}
| r = \frac 1 3 \ln \size {x^3 + a^3} + C
| c = Primitive of $\dfrac {x^{n - 1} } {\pa... | :$\ds \int \frac {x^2 \rd x} {x^3 + a^3} = \frac 1 3 \ln \size {x^3 + a^3} + C$ | From [[Primitive of Power of x less one over Power of x plus Power of a]]:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^3 + a^3}
| r = \frac 1 3 \ln \size {x^3 + a^3} + C
| c = [[Primitive of Power of x less o... | Primitive of x squared over x cubed plus a cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed/Proof_2 | [
"Primitive of x squared over x cubed plus a cubed",
"Primitives involving x cubed plus a cubed"
] | [] | [
"Primitive of Power of x less one over Power of x plus Power of a",
"Primitive of Power of x less one over Power of x plus Power of a"
] |
proofwiki-9464 | Primitive of Reciprocal of x by x cubed plus a cubed | :$\ds \int \frac {\d x} {x \paren {x^3 + a^3} } = \frac 1 {3 a^3} \ln \size {\frac {x^3} {x^3 + a^3} } + C$ | From Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
Setting $n = 3$:
:$\ds \int \frac {\d x} {x \paren {x^3 + a^3} } = \frac 1 {3 a^3} \ln \size {\frac {x^3} {x^3 + a^3} } + C$
directly.
{{qed}} | :$\ds \int \frac {\d x} {x \paren {x^3 + a^3} } = \frac 1 {3 a^3} \ln \size {\frac {x^3} {x^3 + a^3} } + C$ | From [[Primitive of Reciprocal of x by Power of x plus Power of a|Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$]]:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
Setting $n = 3$:
:$\ds \int \frac {\d x} {x \paren {x^3 + a^3} } = \frac 1 {3 a^3} \ln \size... | Primitive of Reciprocal of x by x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_cubed_plus_a_cubed | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Primitive of Reciprocal of x by Power of x plus Power of a"
] |
proofwiki-9465 | Primitive of Reciprocal of x squared by x cubed plus a cubed | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {6 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } - \frac 1 {a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | First a lemma: | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3} } = \frac {-1} {a^3 x} - \frac 1 {6 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } - \frac 1 {a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | First a [[Definition:Lemma|lemma]]: | Primitive of Reciprocal of x squared by x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed | [
"Primitives involving x cubed plus a cubed",
"Primitive of Reciprocal of x squared by x cubed plus a cubed"
] | [] | [
"Definition:Lemma"
] |
proofwiki-9466 | Primitive of Reciprocal of x cubed plus a cubed squared | :$\ds \int \frac {\d x} {\paren {x^3 + a^3}^2} = \frac x {3 a^3 \paren {x^3 + a^3} } + \frac 1 {9 a^5} \map \ln {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 2 {3 a^5 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = z
| r = x^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 3 x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {x^3 + a^3}^2}
| r = \int \frac {\d z} {3 x^2 \paren {z + a^3}^2}
| c = Integration by Substitution
}}... | :$\ds \int \frac {\d x} {\paren {x^3 + a^3}^2} = \frac x {3 a^3 \paren {x^3 + a^3} } + \frac 1 {9 a^5} \map \ln {\frac {\paren {x + a}^2} {x^2 - a x + a^2} } + \frac 2 {3 a^5 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = z
| r = x^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 3 x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {x^3 + a^3}^2}
| r = \int \frac {\d z} {3 x^2 \paren {z + a^3}^2}
| c = [[Integration by Substitution]... | Primitive of Reciprocal of x cubed plus a cubed squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_plus_a_cubed_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_plus_a_cubed_squared | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of Power of a x + b by Power of p x + q",
"Primitive of Reciprocal of Power of a x + b by Power of p x + q",
"Primitive of Reciprocal of x cubed plus a cubed"
] |
proofwiki-9467 | Primitive of x over x cubed plus a cubed squared | :$\ds \int \frac {x \rd x} {\paren {x^3 + a^3}^2} = \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {18 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | First a lemma: | :$\ds \int \frac {x \rd x} {\paren {x^3 + a^3}^2} = \frac {x^2} {3 a^3 \paren {x^3 + a^3} } + \frac 1 {18 a^4} \map \ln {\frac {x^2 - a x + a^2} {\paren {x + a}^2} } + \frac 1 {3 a^4 \sqrt 3} \arctan \frac {2 x - a} {a \sqrt 3}$ | First a [[Definition:Lemma|lemma]]: | Primitive of x over x cubed plus a cubed squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_cubed_plus_a_cubed_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_cubed_plus_a_cubed_squared | [
"Primitives involving x cubed plus a cubed",
"Primitive of x over x cubed plus a cubed squared"
] | [] | [
"Definition:Lemma"
] |
proofwiki-9468 | Primitive of x squared over x cubed plus a cubed squared | :$\ds \int \frac {x^2 \rd x} {\paren {x^3 + a^3}^2} = \frac {-1} {3 \paren {x^3 + a^3} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^3 + a^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 3 x^2
| c = Derivative of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^2 \rd x} {\paren {x^3 + a^3}^2}
| r = \int \frac {\d z} {3 z^2}
| c = Integration by Subs... | :$\ds \int \frac {x^2 \rd x} {\paren {x^3 + a^3}^2} = \frac {-1} {3 \paren {x^3 + a^3} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^3 + a^3
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 3 x^2
| c = [[Derivative of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^2 \rd x} {\paren {x^3 + a^3}^2}
| r = \int \frac {\d z} {3 z^2}
| c = [[Integration b... | Primitive of x squared over x cubed plus a cubed squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_cubed_plus_a_cubed_squared | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9469 | Primitive of Reciprocal of x by x cubed plus a cubed squared | :$\ds \int \frac {\d x} {x \paren {x^3 + a^3}^2} = \frac 1 {3 a^3 \paren {x^3 + a^3} } + \frac 1 {3 a^6} \ln \size {\frac {x^3} {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^3 + a^3}^2}
| r = \int \frac {a^3 \rd x} {a^3 x \paren {x^3 + a^3}^2}
| c = multiplying top and bottom by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x \paren {x^3 + a^3}^2}
| c =
}}
{{eqn | r = \frac 1 {a^3} \int \fra... | :$\ds \int \frac {\d x} {x \paren {x^3 + a^3}^2} = \frac 1 {3 a^3 \paren {x^3 + a^3} } + \frac 1 {3 a^6} \ln \size {\frac {x^3} {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^3 + a^3}^2}
| r = \int \frac {a^3 \rd x} {a^3 x \paren {x^3 + a^3}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x \paren {x^3 + a^3}^2}... | Primitive of Reciprocal of x by x cubed plus a cubed squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_cubed_plus_a_cubed_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_cubed_plus_a_cubed_squared | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x by x cubed plus a cubed",
"Primitive of x squared over x cubed plus a cubed squared"
] |
proofwiki-9470 | Primitive of Reciprocal of x squared by x cubed plus a cubed squared | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3}^2} = \frac {-1} {a^6 x} - \frac {x^2} {3 a^6 \paren {x^3 + a^3} } - \frac 4 {3 a^6} \int \frac {x \rd x} {x^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^3 + a^3}^2}
| r = \int \frac {a^3 \rd x} {a^3 x^2 \paren {x^3 + a^3}^2}
| c = multiplying top and bottom by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^2 \paren {x^3 + a^3}^2}
| c =
}}
{{eqn | r = \frac 1 {a^3} \in... | :$\ds \int \frac {\d x} {x^2 \paren {x^3 + a^3}^2} = \frac {-1} {a^6 x} - \frac {x^2} {3 a^6 \paren {x^3 + a^3} } - \frac 4 {3 a^6} \int \frac {x \rd x} {x^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^3 + a^3}^2}
| r = \int \frac {a^3 \rd x} {a^3 x^2 \paren {x^3 + a^3}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^3$
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^2 \paren {x^3 + a... | Primitive of Reciprocal of x squared by x cubed plus a cubed squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_cubed_plus_a_cubed_squared | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by x cubed plus a cubed/Lemma",
"Primitive of x over x cubed plus a cubed squared/Lemma"
] |
proofwiki-9471 | Primitive of Power of x over x cubed plus a cubed | :$\ds \int \frac {x^m \rd x} {x^3 + a^3} = \frac {x^{m - 2} } {m - 2} - a^3 \int \frac {x^{m - 3} \rd x} {x^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {x^3 + a^3}
| r = \int \frac {x^{m - 3} \paren {x^3} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {x^{m - 3} \paren {x^3 + a^3 - a^3} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {x^{m - 3} \paren {x^3 + a^3} \rd x} {x^3 + a^3} - a^3 \int ... | :$\ds \int \frac {x^m \rd x} {x^3 + a^3} = \frac {x^{m - 2} } {m - 2} - a^3 \int \frac {x^{m - 3} \rd x} {x^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {x^3 + a^3}
| r = \int \frac {x^{m - 3} \paren {x^3} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {x^{m - 3} \paren {x^3 + a^3 - a^3} \rd x} {x^3 + a^3}
| c =
}}
{{eqn | r = \int \frac {x^{m - 3} \paren {x^3 + a^3} \rd x} {x^3 + a^3} - a^3 \int ... | Primitive of Power of x over x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_x_cubed_plus_a_cubed | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9472 | Primitive of Reciprocal of Power of x by x cubed plus a cubed | :$\ds \int \frac {\d x} {x^n \paren {x^3 + a^3} } = \frac {-1} {a^3 \paren {n - 1} x^{n - 1} } - \frac 1 {a^3} \int \frac {\d x} {x^{n - 3} \paren {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^n \paren {x^3 + a^3} }
| r = \int \frac {a^3 \rd x} {a^3 x^n \paren {x^3 + a^3} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^n \paren {x^3 + a^3} }
| c =
}}
{{eqn | r = \frac 1 {a^3} \int \frac {\paren {x^3 + a^3} \rd x} {x^... | :$\ds \int \frac {\d x} {x^n \paren {x^3 + a^3} } = \frac {-1} {a^3 \paren {n - 1} x^{n - 1} } - \frac 1 {a^3} \int \frac {\d x} {x^{n - 3} \paren {x^3 + a^3} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^n \paren {x^3 + a^3} }
| r = \int \frac {a^3 \rd x} {a^3 x^n \paren {x^3 + a^3} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^3 + a^3 - x^3} \rd x} {a^3 x^n \paren {x^3 + a^3} }
| c =
}}
{{eqn | r = \frac 1 {a^3} \int \frac {\paren {x^3 + a^3} \rd x} {x^... | Primitive of Reciprocal of Power of x by x cubed plus a cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_x_cubed_plus_a_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_x_cubed_plus_a_cubed | [
"Primitives involving x cubed plus a cubed"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9473 | Primitive of Reciprocal of x fourth plus a fourth | :$\ds \int \frac {\d x} {x^4 + a^4} = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \int \paren {\frac {x + a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } - \frac {x - a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | n = 1
| r = \frac 1 {4 a^3 \s... | :$\ds \int \frac {\d x} {x^4 + a^4} = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \int \paren {\frac {x + a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } - \frac {x - a \sqrt 2} {2 a^3 \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x fourth plus a fourth/Partial Fraction Expa... | Primitive of Reciprocal of x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Primitive of Reciprocal of x fourth plus a fourth/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function under its Derivative",
"Primitive of Reciprocal of x fourth plus a fourth/Lemma 1",
"Linear Combination of Inte... |
proofwiki-9474 | Primitive of x over x fourth plus a fourth | :$\ds \int \frac {x \rd x} {x^4 + a^4} = \frac 1 {2 a^2} \arctan \frac {x^2} {a^2}$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^4 + a^4}
| r = \int \frac {\d z} {2 \paren {z^2 + \paren {a^2}^2} }
| c = Integration by Substitution
}}
{... | :$\ds \int \frac {x \rd x} {x^4 + a^4} = \frac 1 {2 a^2} \arctan \frac {x^2} {a^2}$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^4 + a^4}
| r = \int \frac {\d z} {2 \paren {z^2 + \paren {a^2}^2} }
| c = [[Integration by Substitution]]
... | Primitive of x over x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_x_over_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_over_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9475 | Primitive of x squared over x fourth plus a fourth | :$\ds \int \frac {x^2 \rd x} {x^4 + a^4} = \frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^4 + a^4}
| r = \int \paren {\frac x {2 a \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } - \frac x {2 a \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | n = 1
| r = \frac 1 {4 a \sqrt 2} \int \frac {2 x \rd x}... | :$\ds \int \frac {x^2 \rd x} {x^4 + a^4} = \frac 1 {4 a \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } - \frac 1 {2 a \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^4 + a^4}
| r = \int \paren {\frac x {2 a \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } - \frac x {2 a \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} } } \rd x
| c = [[Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion|Partial Fraction Exp... | Primitive of x squared over x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function under its Derivative",
"Primitive of Reciprocal of x fourth plus a fourth/Lemma 2",
"Linear Combination of Int... |
proofwiki-9476 | Primitive of x cubed over x fourth plus a fourth | :$\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^4
| r = 4 x^3
| c = Primitive of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^3 \rd x} {x^4 + a^4}
| r = \frac 1 4 \ln \size {x^4 + a^4} + C
| c = Primitive of Function under its Derivative
}}
{{eqn | r = \frac {\map \ln {x^4 + a^4} } 4 + C
... | :$\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^4
| r = 4 x^3
| c = [[Primitive of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^3 \rd x} {x^4 + a^4}
| r = \frac 1 4 \ln \size {x^4 + a^4} + C
| c = [[Primitive of Function under its Derivative]]
}}
{{eqn | r = \frac {\map \ln {x^4 + a^4} ... | Primitive of x cubed over x fourth plus a fourth/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_plus_a_fourth/Proof_1 | [
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of x cubed over x fourth plus a fourth"
] | [] | [
"Primitive of Power",
"Primitive of Function under its Derivative",
"Absolute Value of Even Power"
] |
proofwiki-9477 | Primitive of x cubed over x fourth plus a fourth | :$\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$ | From Primitive of $\dfrac {x^{n - 1} } {x^n + a^n}$:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^4 + a^4}
| r = \frac 1 4 \ln \size {x^4 + a^4} + C
| c = Primitive of $\dfrac {x^{n - 1} } {\paren {x^n + a^n} }$... | :$\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$ | From [[Primitive of Power of x less one over Power of x plus Power of a|Primitive of $\dfrac {x^{n - 1} } {x^n + a^n}$]]:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^4 + a^4}
| r = \frac 1 4 \ln \size {x^4 + a^4} +... | Primitive of x cubed over x fourth plus a fourth/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_plus_a_fourth/Proof_2 | [
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of x cubed over x fourth plus a fourth"
] | [] | [
"Primitive of Power of x less one over Power of x plus Power of a",
"Primitive of Power of x less one over Power of x plus Power of a",
"Absolute Value of Even Power"
] |
proofwiki-9478 | Primitive of Reciprocal of x by x fourth plus a fourth | :$\ds \int \frac {\d x} {x \paren {x^4 + a^4} } = \frac 1 {4 a^4} \map \ln {\frac {x^4} {x^4 + a^4} }$ | From Primitive of Reciprocal of x by Power of x plus Power of a:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^4 + a^4} }
| r = \frac 1 {4 a^4} \ln \size {\frac {x^4} {x^4 + a^4} } + C
... | :$\ds \int \frac {\d x} {x \paren {x^4 + a^4} } = \frac 1 {4 a^4} \map \ln {\frac {x^4} {x^4 + a^4} }$ | From [[Primitive of Reciprocal of x by Power of x plus Power of a]]:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^4 + a^4} }
| r = \frac 1 {4 a^4} \ln \size {\frac {x^4} {x^4 + a^4} } + C
... | Primitive of Reciprocal of x by x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Absolute Value of Even Power"
] |
proofwiki-9479 | Primitive of Reciprocal of x squared by x fourth plus a fourth | :$\ds \int \frac {\d x} {x^2 \paren {x^4 + a^4} } = \frac {-1} {a^4 x} - \frac {-1} {4 a^5 \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } + \frac 1 {2 a^5 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x^2 \paren {x^4 + a^4} }
| o =
| c =
}}
{{eqn | r = \int \frac {a^4 \rd x} {a^4 x^2 \paren {x^4 + a^4} }
| c = multiplying top and bottom by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^2 \paren {x^4 + a^4} }
| c =
}}
{{e... | :$\ds \int \frac {\d x} {x^2 \paren {x^4 + a^4} } = \frac {-1} {a^4 x} - \frac {-1} {4 a^5 \sqrt 2} \map \ln {\frac {x^2 - a x \sqrt 2 + a^2} {x^2 + a x \sqrt 2 + a^2} } + \frac 1 {2 a^5 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arctan {1 + \frac {x \sqrt 2} a} }$ | {{begin-eqn}}
{{eqn | r = \int \frac {\d x} {x^2 \paren {x^4 + a^4} }
| o =
| c =
}}
{{eqn | r = \int \frac {a^4 \rd x} {a^4 x^2 \paren {x^4 + a^4} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd ... | Primitive of Reciprocal of x squared by x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of x squared over x fourth plus a fourth"
] |
proofwiki-9480 | Primitive of Reciprocal of x cubed by x fourth plus a fourth | :$\ds \int \frac {\d x} {x^3 \paren {x^4 + a^4} } = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^6} \arctan \frac {x^2} {a^2}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^4 + a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^3 \paren {x^4 + a^4} }
| c = multiplying top and bottom by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^3 \paren {x^4 + a^4} }
| c =
}}
{{eqn | r = \frac 1 {a^4} \int \... | :$\ds \int \frac {\d x} {x^3 \paren {x^4 + a^4} } = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^6} \arctan \frac {x^2} {a^2}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^4 + a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^3 \paren {x^4 + a^4} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^3 \paren {x^4 + a^4... | Primitive of Reciprocal of x cubed by x fourth plus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_fourth_plus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_fourth_plus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of x over x fourth plus a fourth"
] |
proofwiki-9481 | Primitive of Reciprocal of x fourth minus a fourth | :$\ds \int \frac {\d x} {x^4 - a^4} = \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 - a^4}
| r = \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {a^2 \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = multiplying top and bottom by $a^2$
}}
{{eqn | r = \int \fra... | :$\ds \int \frac {\d x} {x^4 - a^4} = \frac 1 {4 a^3} \ln \size {\frac {x - a} {x + a} } - \frac 1 {2 a^3} \arctan \frac x a$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 - a^4}
| r = \int \frac {\d x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {a^2 \rd x} {a^2 \paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definitio... | Primitive of Reciprocal of x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Difference of Two Squares",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Difference of Two Squares",
"Linear Combination of Integrals/Indefinite",
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared",... |
proofwiki-9482 | Primitive of x over x fourth minus a fourth | :$\ds \int \frac {x \rd x} {x^4 - a^4} = \frac 1 {4 a^2} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^4 - a^4}
| r = \int \frac {\d z} {2 \paren {z^2 - \paren {a^2}^2} }
| c = Integration by Substitution
}}
{... | :$\ds \int \frac {x \rd x} {x^4 - a^4} = \frac 1 {4 a^2} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^4 - a^4}
| r = \int \frac {\d z} {2 \paren {z^2 - \paren {a^2}^2} }
| c = [[Integration by Substitution]]
... | Primitive of x over x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_x_over_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_over_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9483 | Primitive of x squared over x fourth minus a fourth | :$\ds \int \frac {x^2 \rd x} {x^4 - a^4} = \frac 1 {4 a} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^4 - a^4}
| r = \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c =
}}
{{eqn | r = \int \frac {\pare... | :$\ds \int \frac {x^2 \rd x} {x^4 - a^4} = \frac 1 {4 a} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^4 - a^4}
| r = \int \frac {x^2 \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - a^2} \rd x} {\paren {x^2 + a^2} \paren {x^2 - a^2} }
| c =
}}
{{eqn | r = \int \frac {\... | Primitive of x squared over x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Difference of Two Squares",
"Linear Combination of Integrals/Indefinite",
"Difference of Two Squares",
"Linear Combination of Integrals/Indefinite",
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"... |
proofwiki-9484 | Primitive of x cubed over x fourth minus a fourth | :$\ds \int \frac {x^3 \rd x} {x^4 - a^4} = \frac {\ln \size {x^4 - a^4} } 4 + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^4
| r = 4 x^3
| c = Primitive of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^3 \rd x} {x^4 - a^4}
| r = \frac {\ln \size {x^4 - a^4} } 4 + C
| c = Primitive of Function under its Derivative
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {x^3 \rd x} {x^4 - a^4} = \frac {\ln \size {x^4 - a^4} } 4 + C$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^4
| r = 4 x^3
| c = [[Primitive of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^3 \rd x} {x^4 - a^4}
| r = \frac {\ln \size {x^4 - a^4} } 4 + C
| c = [[Primitive of Function under its Derivative]]
}}
{{end-eqn}}
{{qed}} | Primitive of x cubed over x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Primitive of Power",
"Primitive of Function under its Derivative"
] |
proofwiki-9485 | Primitive of Reciprocal of x by x fourth minus a fourth | :$\ds \int \frac {\d x} {x \paren {x^4 - a^4} } = \frac 1 {4 a^4} {\ln \size {\frac {x^4 - a^4} {x^4} } } + C$ | From Primitive of $\dfrac 1 {x \paren {x^n - a^n} }$:
:$\ds \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$
Setting $n = 4$:
:$\ds \int \frac {\d x} {x \paren {x^4 - a^4} } = \frac 1 {4 a^4} \ln \size {\frac {x^4 - a^4} {x^4} } + C$
directly.
{{qed}} | :$\ds \int \frac {\d x} {x \paren {x^4 - a^4} } = \frac 1 {4 a^4} {\ln \size {\frac {x^4 - a^4} {x^4} } } + C$ | From [[Primitive of Reciprocal of x by Power of x minus Power of a|Primitive of $\dfrac 1 {x \paren {x^n - a^n} }$]]:
:$\ds \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$
Setting $n = 4$:
:$\ds \int \frac {\d x} {x \paren {x^4 - a^4} } = \frac 1 {4 a^4} \ln \siz... | Primitive of Reciprocal of x by x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Primitive of Reciprocal of x by Power of x minus Power of a"
] |
proofwiki-9486 | Primitive of Reciprocal of x squared by x fourth minus a fourth | :$\ds \int \frac {\d x} {x^2 \paren {x^4 - a^4} } = \frac 1 {a^4 x} + \frac 1 {4 a^5} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a^5} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^4 - a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^2 \paren {x^4 - a^4} }
| c = multiplying top and bottom by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^2 \paren {x^4 - a^4} }
| c =
}}
{{eqn | r = \int \frac {\paren {... | :$\ds \int \frac {\d x} {x^2 \paren {x^4 - a^4} } = \frac 1 {a^4 x} + \frac 1 {4 a^5} \ln \size {\frac {x - a} {x + a} } + \frac 1 {2 a^5} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^4 - a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^2 \paren {x^4 - a^4} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^2 \paren {x^4 - a^4... | Primitive of Reciprocal of x squared by x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of x squared over x fourth minus a fourth"
] |
proofwiki-9487 | Primitive of Reciprocal of x cubed by x fourth minus a fourth | :$\ds \int \frac {\d x} {x^3 \paren {x^4 - a^4} } = \frac 1 {2 a^4 x^2} + \frac 1 {4 a^6} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^4 - a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^3 \paren {x^4 - a^4} }
| c = multiplying top and bottom by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^3 \paren {x^4 - a^4} }
| c =
}}
{{eqn | r = \int \frac {\paren {... | :$\ds \int \frac {\d x} {x^3 \paren {x^4 - a^4} } = \frac 1 {2 a^4 x^2} + \frac 1 {4 a^6} \ln \size {\frac {x^2 - a^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^4 - a^4} }
| r = \int \frac {a^4 \rd x} {a^4 x^3 \paren {x^4 - a^4} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^4$
}}
{{eqn | r = \int \frac {\paren {x^4 + a^4 - x^4} \rd x} {a^4 x^3 \paren {x^4 - a^4... | Primitive of Reciprocal of x cubed by x fourth minus a fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_fourth_minus_a_fourth | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_fourth_minus_a_fourth | [
"Primitives involving x to the fourth plus or minus a to the fourth"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of x over x fourth minus a fourth"
] |
proofwiki-9488 | Primitive of Reciprocal of x by Power of x plus Power of a | :$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n + a^n} }
| r = \int \frac {a^n \rd x} {a^n x \paren {x^2 + a^2} }
| c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x \paren {x^n + a^n} }
| c = adding and subtracting $x^n$
}}
{{eqn | r ... | :$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n + a^n} }
| r = \int \frac {a^n \rd x} {a^n x \paren {x^2 + a^2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x \paren {x^n + a^n} }
... | Primitive of Reciprocal of x by Power of x plus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_plus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power of x less one over Power of x plus Power of a",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9489 | Absolute Value of Even Power | Let $x \in \R$ be a real number.
Let $n \in \Z$ be an integer.
Then:
:$\size {x^{2 n} } = x^{2 n}$ | From Even Power is Non-Negative:
:$x^{2 n} \ge 0$
The result follows from the definition of absolute value.
{{qed}}
Category:Absolute Value Function
4q0q24ahaqcyqiyl1er3fltsy5q6dxm | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\size {x^{2 n} } = x^{2 n}$ | From [[Even Power is Non-Negative]]:
:$x^{2 n} \ge 0$
The result follows from the definition of [[Definition:Absolute Value|absolute value]].
{{qed}}
[[Category:Absolute Value Function]]
4q0q24ahaqcyqiyl1er3fltsy5q6dxm | Absolute Value of Even Power | https://proofwiki.org/wiki/Absolute_Value_of_Even_Power | https://proofwiki.org/wiki/Absolute_Value_of_Even_Power | [
"Absolute Value Function"
] | [
"Definition:Real Number",
"Definition:Integer"
] | [
"Even Power is Non-Negative",
"Definition:Absolute Value",
"Category:Absolute Value Function"
] |
proofwiki-9490 | Primitive of Power of x less one over Power of x plus Power of a | :$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^n + a^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = Power Rule for Derivatives and Derivative of Constant
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^{n - 1} \rd x} {x^n + a^n}
| r = \frac 1 n \ln \size {... | :$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^n + a^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = [[Power Rule for Derivatives]] and [[Derivative of Constant]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^{n - 1} \rd x} {x^n + a^n}
| r = \frac 1 n \ln... | Primitive of Power of x less one over Power of x plus Power of a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_less_one_over_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_less_one_over_Power_of_x_plus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Primitive of Function under its Derivative"
] |
proofwiki-9491 | Primitive of Power of x over Power of Power of x plus Power of a | :$\ds \int \frac {x^m \rd x} {\paren {x^n + a^n}^r} = \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^{r - 1} } - a^n \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^r}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n + a^n}^r}
| r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n - a^n} \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n} \rd x}... | :$\ds \int \frac {x^m \rd x} {\paren {x^n + a^n}^r} = \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^{r - 1} } - a^n \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^r}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n + a^n}^r}
| r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n - a^n} \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n} \rd x}... | Primitive of Power of x over Power of Power of x plus Power of a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_Power_of_x_plus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9492 | Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a | :$\ds \int \frac {\d x} {x^m \paren {x^n + a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^n + a^n}^r}
| r = \int \frac {a^n \rd x} {a^n x^m \paren {x^n + a^n}^r}
| c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x^m \paren {x^n + a^n}^r}
| c = adding and subtracting $x^n$
}}
{... | :$\ds \int \frac {\d x} {x^m \paren {x^n + a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^n + a^n}^r}
| r = \int \frac {a^n \rd x} {a^n x^m \paren {x^n + a^n}^r}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x^m \paren {x^n + a... | Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_Power_of_x_plus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9493 | Primitive of Reciprocal of x by Root of Power of x plus Power of a | :$\ds \int \frac {\d x} {x \sqrt {x^n + a^n} } = \frac 1 {n \sqrt {a^n} } \ln \size {\frac {\sqrt {x^n + a^n} - \sqrt {a^n} } {\sqrt {x^n + a^n} + \sqrt {a^n} } } + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {x^n + a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {n x^{n - 1} } {2 \sqrt {x^n + a^n} }
| c = Derivative of Power, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {x^n + a^n} }... | :$\ds \int \frac {\d x} {x \sqrt {x^n + a^n} } = \frac 1 {n \sqrt {a^n} } \ln \size {\frac {\sqrt {x^n + a^n} - \sqrt {a^n} } {\sqrt {x^n + a^n} + \sqrt {a^n} } } + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {x^n + a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {n x^{n - 1} } {2 \sqrt {x^n + a^n} }
| c = [[Derivative of Power]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {x^n ... | Primitive of Reciprocal of x by Root of Power of x plus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_Power_of_x_plus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared minus a squared"
] |
proofwiki-9494 | Primitive of Reciprocal of x by Power of x minus Power of a | :$\ds \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n - a^n} }
| r = \int \frac {a^n \rd x} {a^n x \paren {x^2 - a^2} }
| c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {-\paren {x^n - a^n} + x^n} \rd x} {a^n x \paren {x^n - a^n} }
| c = adding and subtracting $x^n$
}}
... | :$\ds \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n - a^n} }
| r = \int \frac {a^n \rd x} {a^n x \paren {x^2 - a^2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^n$
}}
{{eqn | r = \int \frac {\paren {-\paren {x^n - a^n} + x^n} \rd x} {a^n x \paren {x^n -... | Primitive of Reciprocal of x by Power of x minus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_minus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_minus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power of x less one over Power of x minus Power of a",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9495 | Primitive of Power of x less one over Power of x minus Power of a | :$\ds \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^n - a^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = Power Rule for Derivatives and Derivative of Constant
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^{n - 1} \rd x} {x^n - a^n}
| r = \frac 1 n \ln \size {... | :$\ds \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^n - a^n
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = n x^{n - 1}
| c = [[Power Rule for Derivatives]] and [[Derivative of Constant]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x^{n - 1} \rd x} {x^n - a^n}
| r = \frac 1 n \ln... | Primitive of Power of x less one over Power of x minus Power of a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_less_one_over_Power_of_x_minus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_less_one_over_Power_of_x_minus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Primitive of Function under its Derivative"
] |
proofwiki-9496 | Primitive of Power of x over Power of Power of x minus Power of a | :$\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r} = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n - a^n}^r}
| r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n + a^n} \rd x} {\paren {x^n - a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n} \rd x}... | :$\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r} = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n - a^n}^r}
| r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n + a^n} \rd x} {\paren {x^n - a^n}^r}
| c =
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n} \rd x}... | Primitive of Power of x over Power of Power of x minus Power of a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_Power_of_x_minus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_Power_of_x_minus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9497 | Primitive of Reciprocal of Power of x by Power of Power of x minus Power of a | :$\ds \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n - a^n}^r} - \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n - a^n}^{r - 1} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r}
| r = \int \frac {a^n \rd x} {a^n x^m \ \paren {x^n - a^n}^r}
| c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n - \paren {x^n - a^n} } \rd x} {a^n x^m \ \paren {x^n + a^n}^r}
| c = adding and subtra... | :$\ds \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n - a^n}^r} - \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n - a^n}^{r - 1} }$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r}
| r = \int \frac {a^n \rd x} {a^n x^m \ \paren {x^n - a^n}^r}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n - \paren {x^n - a^n} } \rd x} {a^n x^m \... | Primitive of Reciprocal of Power of x by Power of Power of x minus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_Power_of_x_minus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_Power_of_x_minus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9498 | Primitive of Reciprocal of x by Root of Power of x minus Power of a | :$\ds \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$ | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {x^n - a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {n x^{n - 1} } {2 \sqrt {x^n - a^n} }
| c = Derivative of Power, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {x^n - a^n} }... | :$\ds \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$ | {{begin-eqn}}
{{eqn | l = u
| r = \sqrt {x^n - a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {n x^{n - 1} } {2 \sqrt {x^n - a^n} }
| c = [[Derivative of Power]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \sqrt {x^n ... | Primitive of Reciprocal of x by Root of Power of x minus Power of a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_Power_of_x_minus_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_Power_of_x_minus_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9499 | Primitive of Power of x over Even Power of x minus Even Power of a | {{begin-eqn}}
{{eqn | l = \int \frac {x^{p - 1} \rd x} {x^{2 m} - a^{2 m} }
| r = \frac 1 {2 m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \cos {\frac {k p \pi} m} \map \ln {x^2 - 2 a x \map \cos {\frac {k \pi} m} + a^2}
}}
{{eqn | o =
| ro= -
| r = \frac 1 {m a^{2 m - p} } \sum_{k \mathop = 1}^{m... | The integrand is a rational function.
It has simple poles at $x = \omega_k a$ where $\omega_k = e^{\pi i k /m} $, $k = 0, 1, \ldots, 2 m - 1$ are the $2m$'th roots of unity:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {x^{p - 1} } {x^{2 m} - a^{2 m} }
| c =
}}
{{eqn | r = \dfrac {x^{p - 1} } {\ds \prod... | {{begin-eqn}}
{{eqn | l = \int \frac {x^{p - 1} \rd x} {x^{2 m} - a^{2 m} }
| r = \frac 1 {2 m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \cos {\frac {k p \pi} m} \map \ln {x^2 - 2 a x \map \cos {\frac {k \pi} m} + a^2}
}}
{{eqn | o =
| ro= -
| r = \frac 1 {m a^{2 m - p} } \sum_{k \mathop = 1}^{m... | The [[Definition:Integrand|integrand]] is a [[Definition:Rational Function|rational function]].
It has [[Definition:Simple Pole|simple poles]] at $x = \omega_k a$ where $\omega_k = e^{\pi i k /m} $, $k = 0, 1, \ldots, 2 m - 1$ are the [[Definition:Complex Roots of Unity|$2m$'th roots of unity]]:
{{begin-eqn}}
{{eqn |... | Primitive of Power of x over Even Power of x minus Even Power of a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Even_Power_of_x_minus_Even_Power_of_a | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Even_Power_of_x_minus_Even_Power_of_a | [
"Primitives involving Power of x plus or minus Power of a"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Rational Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Root of Unity/Complex",
"Definition:Residue (Complex Analysis)",
"Definition:Primitive (Calculus)"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.