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-9300 | Primitive of x over a squared minus x squared | :$\ds \int \frac {x \rd x} {a^2 - x^2} = -\frac 1 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {a^2 - x^2} }
| r = \int \frac {\d z} {-2 z}
| c = Integration b... | :$\ds \int \frac {x \rd x} {a^2 - x^2} = -\frac 1 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {a^2 - x^2} }
| r = \int \frac {\d z} {-2 z}
| c = [[Integra... | Primitive of x over a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal/Corollary"
] |
proofwiki-9301 | Linear Combination of Laplace Transforms | Then:
:$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$
everywhere all the above expressions are defined. | {{begin-eqn}}
{{eqn | l = \laptrans {\lambda \, \map f t + \mu \, \map g t}
| r = \int_0^{\to +\infty} e^{-s t} \paren {\lambda \, \map f t + \mu \, \map g t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{A \mathop \to +\infty} \paren {\int_0^A e^{-s t} \paren {\lambda \, \map f t + \mu \, \m... | Then:
:$\laptrans {\lambda \, \map f t + \mu \, \map g t} = \lambda \laptrans {\map f t} + \mu \laptrans {\map g t}$
everywhere all the above expressions are defined. | {{begin-eqn}}
{{eqn | l = \laptrans {\lambda \, \map f t + \mu \, \map g t}
| r = \int_0^{\to +\infty} e^{-s t} \paren {\lambda \, \map f t + \mu \, \map g t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{A \mathop \to +\infty} \paren {\int_0^A e^{-s t} \paren {\lambda \, \map f t + \mu \, \m... | Linear Combination of Laplace Transforms | https://proofwiki.org/wiki/Linear_Combination_of_Laplace_Transforms | https://proofwiki.org/wiki/Linear_Combination_of_Laplace_Transforms | [
"Linear Combinations of Laplace Transforms",
"Properties of Laplace Transforms"
] | [] | [
"Linear Combination of Complex Integrals",
"Combination Theorem for Limits at Infinity"
] |
proofwiki-9302 | Primitive of x cubed over a squared minus x squared | :$\ds \int \frac {x^3 \rd x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {a^2 - x^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-x \paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2 x} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = -\int x \rd x + a^2 \int \frac {x ... | :$\ds \int \frac {x^3 \rd x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {a^2 - x^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-x \paren {a^2 - x^2} } {a^2 - x^2} \rd x + \int \frac {a^2 x} {a^2 - x^2} \rd x
| c =
}}
{{eqn | r = -\int x \rd x + a^2 \int \frac {x ... | Primitive of x cubed over a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Power",
"Primitive of x over a squared minus x squared"
] |
proofwiki-9303 | Primitive of Reciprocal of x by a squared minus x squared | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 x} + \frac x {a^2 \paren {a^2 - x^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} x + \frac 1 {a^2} \int \frac {x \rd x} {a^2 - x^2}
| c = Linear Combination... | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 x} + \frac x {a^2 \paren {a^2 - x^2} } } \rd x
| c = [[Primitive of Reciprocal of x by a squared minus x squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2} \int \frac... | Primitive of Reciprocal of x by a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x by a squared minus x squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over a squared minus x squared",
"Logarithm of Power",
"Square of Real Number is Non-Negative",
"Difference of Logarithms"
] |
proofwiki-9304 | Primitive of Reciprocal of x squared by a squared minus x squared | :$\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2} } = \frac {-1} {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 \paren {a^2 - x^2} } + \frac 1 {a^2 x^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} {a^2 - x^2} + \frac 1 {a^2} \int \frac {\d x} {x^2}
| c = Linear Combin... | :$\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2} } = \frac {-1} {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 \paren {a^2 - x^2} } + \frac 1 {a^2 x^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by a squared minus x squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2... | Primitive of Reciprocal of x squared by a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x squared by a squared minus x squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form"
] |
proofwiki-9305 | Primitive of Reciprocal of x cubed by a squared minus x squared | :$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2} } = \frac {-1} {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 x^3} + \frac 1 {a^4 x} + \frac x {a^4 \paren {a^2 - x^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} {x^3} + \frac 1 {a^4} \int \frac {\d x} x + \frac 1 {a^4} \... | :$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2} } = \frac {-1} {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a^2 - x^2} }
| r = \int \paren {\frac 1 {a^2 x^3} + \frac 1 {a^4 x} + \frac x {a^4 \paren {a^2 - x^2} } } \rd x
| c = [[Primitive of Reciprocal of x cubed by a squared minus x squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | ... | Primitive of Reciprocal of x cubed by a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x cubed by a squared minus x squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of x over a squared minus x squared",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9306 | Primitive of Reciprocal of a squared minus x squared squared | :$\ds \int \frac {\d x} {\paren {a^2 - x^2}^2} = \frac x {2 a^2 \paren {a^2 - x^2} } + \frac 1 {4 a^3} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {4 a^2} \paren {\dfrac 1 {\paren {a - x}^2} + \frac 1 {\paren {a + x}^2} + \frac 1 {a \paren {a + x} } + \frac 1 {a \paren {a - x} } } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \int \paren {\frac 1 {4... | :$\ds \int \frac {\d x} {\paren {a^2 - x^2}^2} = \frac x {2 a^2 \paren {a^2 - x^2} } + \frac 1 {4 a^3} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {4 a^2} \paren {\dfrac 1 {\paren {a - x}^2} + \frac 1 {\paren {a + x}^2} + \frac 1 {a \paren {a + x} } + \frac 1 {a \paren {a - x} } } } \rd x
| c = [[Primitive of Reciprocal of a squared minus x squared squared/Pa... | Primitive of Reciprocal of a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared_squared | [
"Primitive of Reciprocal of a squared minus x squared squared",
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared squared/Partial Fraction Expansion",
"Reciprocal of Difference of Squares as Sum of Reciprocals",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function of a x + b",
"Primitive of Power",
"Reciprocal of Difference of Squares as Difference... |
proofwiki-9307 | Primitive of x over a squared minus x squared squared | :$\ds \int \frac {x \rd x} {\paren {a^2 - x^2}^2} = \frac 1 {2 \paren {a^2 - x^2} } + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Derivative of Power
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {\d z} {-2 z^2}
| c = Integration by S... | :$\ds \int \frac {x \rd x} {\paren {a^2 - x^2}^2} = \frac 1 {2 \paren {a^2 - x^2} } + C$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Derivative of Power]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {\d z} {-2 z^2}
| c = [[Integratio... | Primitive of x over a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_over_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9308 | Primitive of x squared over a squared minus x squared squared | :$\ds \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2} = \frac x {2 \paren {a^2 - x^2} } - \frac 1 {4 a} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {x^2 - a^2 + a^2} {\paren {a^2 - x^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {\paren {a^2 - x^2}^2} \rd x + a^2 \int \frac {\d x} {\paren {a^2 - x^2}^2}
| c = Linear Combination of Primit... | :$\ds \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2} = \frac x {2 \paren {a^2 - x^2} } - \frac 1 {4 a} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {x^2 - a^2 + a^2} {\paren {a^2 - x^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-\paren {a^2 - x^2} } {\paren {a^2 - x^2}^2} \rd x + a^2 \int \frac {\d x} {\paren {a^2 - x^2}^2}
| c = [[Linear Combination of Prim... | Primitive of x squared over a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form",
"Primitive of Reciprocal of a squared minus x squared squared"
] |
proofwiki-9309 | Primitive of x cubed over a squared minus x squared squared | :$\ds \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2} = \frac {a^2} {2 \paren {a^2 - x^2} } + \frac 1 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {\paren {a^2 - x^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-x \paren {a^2 - x^2} } {\paren {a^2 - x^2}^2} \rd x + a^2 \int \frac {x \rd x} {\paren {a^2 - x^2}^2}
| c = Linear Comb... | :$\ds \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2} = \frac {a^2} {2 \paren {a^2 - x^2} } + \frac 1 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {\paren {a^2 - x^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {-x \paren {a^2 - x^2} } {\paren {a^2 - x^2}^2} \rd x + a^2 \int \frac {x \rd x} {\paren {a^2 - x^2}^2}
| c = [[Linear Co... | Primitive of x cubed over a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over a squared minus x squared",
"Primitive of x over a squared minus x squared squared"
] |
proofwiki-9310 | Primitive of Reciprocal of x by a squared minus x squared squared | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2}^2} = \frac 1 {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x} + \frac x {a^4 \paren {a^2 - x^2} } + \frac x {a^2 \paren {a^2 - x^2}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^4} \int \frac {\d x} x + \frac 1 {a^4} \int \frac {x \rd x} {a^2... | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2}^2} = \frac 1 {2 a^2 \paren {a^2 - x^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x} + \frac x {a^4 \paren {a^2 - x^2} } + \frac x {a^2 \paren {a^2 - x^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x by a squared minus x squared squared/Partial Fraction Expansion|Partial Fraction Expan... | Primitive of Reciprocal of x by a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x by a squared minus x squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over a squared minus x squared",
"Primitive of x over a squared minus x squared squared",
"Logarithm of Power",
"Differ... |
proofwiki-9311 | Primitive of Reciprocal of x squared by a squared minus x squared squared | :$\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {a + x} } + \frac 3 {4 a^5 \paren {a - x} } + \frac 1 {4 a^4 \paren {a + x}^2} + \frac 1 {4 a^4 \paren {a - x}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac ... | :$\ds \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2} = \frac {-1} {a^4 x} + \frac x {2 a^4 \paren {a^2 - x^2} } + \frac 3 {4 a^5} \map \ln {\frac {a + x} {a - x} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {a + x} } + \frac 3 {4 a^5 \paren {a - x} } + \frac 1 {4 a^4 \paren {a + x}^2} + \frac 1 {4 a^4 \paren {a - x}^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by a squa... | Primitive of Reciprocal of x squared by a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x squared by a squared minus x squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Power",
"Difference of Logarithms",
"Sign of Quotient of Factors of Difference of Squares"
... |
proofwiki-9312 | Primitive of Reciprocal of x cubed by a squared minus x squared squared | :$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2}^2} = \frac {-1} {2 a^4 x^2} + \frac 1 {2 a^4 \paren {a^2 - x^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x^3} + \frac 2 {a^6 x} + \frac {2 x} {a^6 \paren {a^2 - x^2} } + \frac x {a^4 \paren {a^2 - x^2}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^4} \int \frac {\d x} {x^3} + \frac 2 {... | :$\ds \int \frac {\d x} {x^3 \paren {a^2 - x^2}^2} = \frac {-1} {2 a^4 x^2} + \frac 1 {2 a^4 \paren {a^2 - x^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a^2 - x^2}^2}
| r = \int \paren {\frac 1 {a^4 x^3} + \frac 2 {a^6 x} + \frac {2 x} {a^6 \paren {a^2 - x^2} } + \frac x {a^4 \paren {a^2 - x^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x cubed by a squared minus x squared squared/Partial Fraction ... | Primitive of Reciprocal of x cubed by a squared minus x squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_squared_minus_x_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_squared_minus_x_squared_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x cubed by a squared minus x squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of x over a squared minus x squared",
"Primitive of x over a squared minus x squared squared",
"... |
proofwiki-9313 | Primitive of Reciprocal of Power of a squared minus x squared | :$\ds \int \frac {\d x} {\paren {a^2 - x^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {a^2 - x^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {a^2 - x^2}^{n - 1} }$
for $x^2 > a^2$. | Aiming for an expression in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \ \frac {\d u} {\d x} \rd x$
in order to use the technique of Integration by Parts, let:
{{begin-eqn}}
{{eqn | l = v
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = 1
| c = Po... | :$\ds \int \frac {\d x} {\paren {a^2 - x^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {a^2 - x^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {a^2 - x^2}^{n - 1} }$
for $x^2 > a^2$. | Aiming for an expression in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \ \frac {\d u} {\d x} \rd x$
in order to use the technique of [[Integration by Parts]], let:
{{begin-eqn}}
{{eqn | l = v
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = 1
| ... | Primitive of Reciprocal of Power of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Integration by Parts",
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integration by Parts",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9314 | Primitive of x over Power of a squared minus x squared | :$\ds \int \frac {x \rd x} {\paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} \paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {a^2 - x^2}^n}
| r = \int \frac {\d z} {- 2 z^n}
| c = Integrat... | :$\ds \int \frac {x \rd x} {\paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} \paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {a^2 - x^2}^n}
| r = \int \frac {\d z} {- 2 z^n}
| c = [[In... | Primitive of x over Power of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9315 | Primitive of Reciprocal of x by Power of a squared minus x squared | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} a^2 \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {a^2 - x^2}^n}
| r = \int \frac {\d z} {-2 x^2 z^n}
| c = Integr... | :$\ds \int \frac {\d x} {x \paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} a^2 \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {a^2 - x^2}^n}
| r = \int \frac {\d z} {-2 x^2 z^n}
| c = [[... | Primitive of Reciprocal of x by Power of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power of x by Power of a x + b/Increment of Power of x"
] |
proofwiki-9316 | Primitive of Power of x over Power of a squared minus x squared | :$\ds \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n} = a^2 \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^n} - \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {a^2 - x^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {a^2 + x^2 - a^2} \rd x} {\paren {a^2 - x^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {a^2 - \p... | :$\ds \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n} = a^2 \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^n} - \int \frac {x^{m - 2} \rd x} {\paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {a^2 - x^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {a^2 - x^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {a^2 + x^2 - a^2} \rd x} {\paren {a^2 - x^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {a^2 - \p... | Primitive of Power of x over Power of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9317 | Primitive of Reciprocal of Power of x by Power of a squared minus x squared | :$\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} }
| r = \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{n - 1} \paren {a^2 - x^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{\paren {n - 1} + 1} }
| c =
}}
{{eqn | r ... | :$\ds \int \frac {\d x} {x^m \paren {a^2 - x^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {a^2 - x^2}^n}$
for $x^2 < a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a^2 - x^2}^{n - 1} }
| r = \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{n - 1} \paren {a^2 - x^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2} \rd x} {x^m \paren {a^2 - x^2}^{\paren {n - 1} + 1} }
| c =
}}
{{eqn | r ... | Primitive of Reciprocal of Power of x by Power of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_a_squared_minus_x_squared | [
"Primitives involving a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9318 | Primitive of x over Root of x squared plus a squared | :$\ds \int \frac {x \rd x} {\sqrt {x^2 + a^2} } = \sqrt {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = x^2 + a^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = 2 x
| c = Chain Rule for Derivatives, Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {x^2 + a^2} }
| r = \int \frac {z \rd... | :$\ds \int \frac {x \rd x} {\sqrt {x^2 + a^2} } = \sqrt {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = x^2 + a^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = 2 x
| c = [[Chain Rule for Derivatives]], [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {x^2 + a^2} }
| r = \int \fra... | Primitive of x over Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant"
] |
proofwiki-9319 | Primitive of x cubed over Root of x squared plus a squared | :$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 + a^2} } = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{end-eqn}}
... | :$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 + a^2} } = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{end-e... | Primitive of x cubed over Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared plus a squared",
"Integration by Parts",
"Primitive of x by Root of x squared plus a squared"
] |
proofwiki-9320 | Primitive of Reciprocal of x squared by Root of x squared plus a squared | :$\ds \int \frac {\d x} {x^2 \sqrt {x^2 + a^2} } = -\frac {\sqrt {x^2 + a^2} } {a^2 x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^2 \sqrt {x^2 + a^2} }
| r = \int \frac {\d z} {2 z \sqrt z \sqrt {z + a^2} }
| c = Integrat... | :$\ds \int \frac {\d x} {x^2 \sqrt {x^2 + a^2} } = -\frac {\sqrt {x^2 + a^2} } {a^2 x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^2 \sqrt {x^2 + a^2} }
| r = \int \frac {\d z} {2 z \sqrt z \sqrt {z + a^2} }
| c = [[In... | Primitive of Reciprocal of x squared by Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of Power of x by Root of a x + b"
] |
proofwiki-9321 | Primitive of Reciprocal of x cubed by Root of x squared plus a squared | :$\ds \int \frac {\d x} {x^3 \sqrt {x^2 + a^2} } = \frac {-\sqrt {x^2 + a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {x^2 + a^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {z + a^2} }
| c = In... | :$\ds \int \frac {\d x} {x^3 \sqrt {x^2 + a^2} } = \frac {-\sqrt {x^2 + a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {x^2 + a^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {z + a^2} }
| c ... | Primitive of Reciprocal of x cubed by Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of x by Root of x squared plus a squared"
] |
proofwiki-9322 | Primitive of x by Root of x squared plus a squared | :$\ds \int x \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {x^2 + a^2} \rd x
| r = \int \frac {\sqrt z \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = Integration by... | :$\ds \int x \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {x^2 + a^2} \rd x
| r = \int \frac {\sqrt z \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = [[Integrat... | Primitive of x by Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b"
] |
proofwiki-9323 | Primitive of x squared by Root of x squared plus a squared | :$\ds \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^2 \sqrt {x^2 + a^2} \rd x
| r = \int \frac {z \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = Integration by Sub... | :$\ds \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^2 \sqrt {x^2 + a^2} \rd x
| r = \int \frac {z \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = [[Integration ... | Primitive of x squared by Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_plus_a_squared | [
"Primitive of x squared by Root of x squared plus a squared",
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b by Root of p x + q",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q"
] |
proofwiki-9324 | Primitive of x cubed by Root of x squared plus a squared | :$\ds \int x^3 \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {x^2 + a^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = Integration ... | :$\ds \int x^3 \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {x^2 + a^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {z + a^2} \rd z} {2 \sqrt z}
| c = [[Integ... | Primitive of x cubed by Root of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_plus_a_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of x by Root of a x + b"
] |
proofwiki-9325 | Primitive of Root of x squared plus a squared over x cubed | :$\ds \int \frac {\sqrt {x^2 + a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } {x^3} \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 z^{3/2} \sqrt z}
| c = I... | :$\ds \int \frac {\sqrt {x^2 + a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 + a^2} } {2 x^2} - \frac 1 {2 a} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 + a^2} } {x^3} \rd x
| r = \int \frac {\sqrt {z + a^2} \rd z} {2 z^{3/2} \sqrt z}
| c... | Primitive of Root of x squared plus a squared over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_over_x_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-9326 | Primitive of Reciprocal of Root of x squared plus a squared cubed | :$\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sec^2 \theta
| c = Derivative of Tangent Function
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a \sec^2 \theta \rd \the... | :$\ds \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac x {a^2 \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sec^2 \theta
| c = [[Derivative of Tangent Function]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a \sec^2 \theta \rd ... | Primitive of Reciprocal of Root of x squared plus a squared cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared_cubed/Proof_1 | [
"Primitives involving Root of x squared plus a squared",
"Primitive of Reciprocal of Root of x squared plus a squared cubed"
] | [] | [
"Derivative of Tangent Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Cosine Function",
"Tangent is Sine divided by Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-9327 | Primitive of x over Root of x squared plus a squared cubed | :$\ds \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {\sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {x \rd z} {2 x z^{3/2} }
| c = Integ... | :$\ds \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {\sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {x \rd z} {2 x z^{3/2} }
| c = [... | Primitive of x over Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9328 | Primitive of x squared over Root of x squared plus a squared cubed | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \map \ln {x + \sqrt {x^2 + a^2} } + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-... | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \frac {-x} {\sqrt {x^2 + a^2} } + \map \ln {x + \sqrt {x^2 + a^2} } + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
and let:
{... | Primitive of x squared over Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared plus a squared cubed",
"Integration by Parts",
"Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-9329 | Primitive of x cubed over Root of x squared plus a squared cubed | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \sqrt {x^2 + a^2} + \frac {a^2} {\sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2 - a^2} \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 +... | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \sqrt {x^2 + a^2} + \frac {a^2} {\sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2 - a^2} \rd x} {\paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 +... | Primitive of x cubed over Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Root of x squared plus a squared",
"Primitive of x over Root of x squared plus a squared cubed"
] |
proofwiki-9330 | Primitive of Reciprocal of x by Root of x squared plus a squared cubed | :$\ds \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3} = \frac 1 {a^2 \sqrt {x^2 + a^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {z + a^2}... | :$\ds \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3} = \frac 1 {a^2 \sqrt {x^2 + a^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {z + ... | Primitive of Reciprocal of x by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x by Power of Root of a x + b",
"Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-9331 | Primitive of Reciprocal of x squared by Root of x squared plus a squared cubed | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-\sqrt {x^2 + a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-\sqrt {x^2 + a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x squared by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by Root of x squared plus a squared",
"Primitive of Reciprocal of Root of x squared plus a squared cubed"
] |
proofwiki-9332 | Primitive of Reciprocal of x cubed by Root of x squared plus a squared cubed | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {x^2 + a^2} } - \frac 3 {2 a^4 \sqrt {x^2 + a^2} } + \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^3 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {x^2 + a^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {x^2 + a^2} } - \frac 3 {2 a^4 \sqrt {x^2 + a^2} } + \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^3 \paren {\sqrt {x^2 + a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x cubed by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x cubed by Root of x squared plus a squared",
"Primitive of Reciprocal of x by Root of x squared plus a squared cubed"
] |
proofwiki-9333 | Primitive of Root of x squared plus a squared cubed | :$\ds \int \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 \sqrt z} \rd x
| c = ... | :$\ds \int \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 \sqrt z} \rd x
| ... | Primitive of Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q over Root of a x + b",
"Primitive of Root of p x + q over Root of a x + b",
"Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form"
] |
proofwiki-9334 | Primitive of x by Root of x squared plus a squared cubed | :$\ds \int x \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {z^{3/2} } 2 \rd z
| c = Integration by Substi... | :$\ds \int x \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {z^{3/2} } 2 \rd z
| c = [[Integration by ... | Primitive of x by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9335 | Primitive of x squared by Root of x squared plus a squared cubed | :$\ds \int x^2 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^5} 6 - \frac {a^2 x \paren {\sqrt {x^2 + a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {z + a^2}^{3/2} } 2 \rd z
| c = In... | :$\ds \int x^2 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^5} 6 - \frac {a^2 x \paren {\sqrt {x^2 + a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {z + a^2}^{3/2} } 2 \rd z
| c ... | Primitive of x squared by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q by Root of a x + b",
"Primitive of Root of x squared plus a squared cubed"
] |
proofwiki-9336 | Primitive of x cubed by Root of x squared plus a squared cubed | :$\ds \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {x^2 + a^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 + a^2 - a^2} \paren {\sqrt {x^2 + a^2} }^3 \rd x
| c = Primitive of Power
}}
{{eqn | r = \int x \paren {x^2 + a^2} \paren {\sqrt... | :$\ds \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {x^2 + a^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 + a^2 - a^2} \paren {\sqrt {x^2 + a^2} }^3 \rd x
| c = [[Primitive of Power]]
}}
{{eqn | r = \int x \paren {x^2 + a^2} \paren {\... | Primitive of x cubed by Root of x squared plus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_plus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_plus_a_squared_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Primitive of Power",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9337 | Primitive of Root of x squared plus a squared cubed over x | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + a^2 \sqrt {x^2 + a^2} - a^3 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z} \rd z
| ... | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + a^2 \sqrt {x^2 + a^2} - a^3 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z} \rd z
... | Primitive of Root of x squared plus a squared cubed over x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of Root of a x + b over x",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of x squared plus a squared"
] |
proofwiki-9338 | Primitive of Root of x squared plus a squared cubed over x squared | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} x + \frac {3 x \sqrt {x^2 + a^2} } 2 + \frac {3 a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z \sqrt z} \r... | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} x + \frac {3 x \sqrt {x^2 + a^2} } 2 + \frac {3 a^2} 2 \map \ln {x + \sqrt {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z \sqrt z... | Primitive of Root of x squared plus a squared cubed over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x_squared | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of x squared plus a squared"
] |
proofwiki-9339 | Primitive of Root of x squared plus a squared cubed over x cubed | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z^2} \rd z
... | :$\ds \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 + a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 + a^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {z + a^2} }^3} {2 z^2} \rd ... | Primitive of Root of x squared plus a squared cubed over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_plus_a_squared_cubed_over_x_cubed | [
"Primitives involving Root of x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of x squared plus a squared over x"
] |
proofwiki-9340 | Primitive of x over Root of x squared minus a squared | :$\ds \int \frac {x \rd x} {\sqrt {x^2 - a^2} } = \sqrt {x^2 - a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = x^2 - a^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = 2 x
| c = Chain Rule for Derivatives, Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {x^2 - a^2} }
| r = \int \frac {z \rd... | :$\ds \int \frac {x \rd x} {\sqrt {x^2 - a^2} } = \sqrt {x^2 - a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = x^2 - a^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = 2 x
| c = [[Chain Rule for Derivatives]], [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {x^2 - a^2} }
| r = \int \fra... | Primitive of x over Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant"
] |
proofwiki-9341 | Primitive of x cubed over Root of x squared minus a squared | :$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{end-eqn}}
... | :$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{end-e... | Primitive of x cubed over Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared minus a squared",
"Integration by Parts",
"Primitive of x by Root of x squared minus a squared"
] |
proofwiki-9342 | Primitive of Reciprocal of x squared by Root of x squared minus a squared | :$\ds \int \frac {\d x} {x^2 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {a^2 x} + C$
for $\size x > a$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^2 - a^2... | :$\ds \int \frac {\d x} {x^2 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {a^2 x} + C$
for $\size x > a$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^... | Primitive of Reciprocal of x squared by Root of x squared minus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_minus_a_squared/Proof_1 | [
"Primitive of Reciprocal of x squared by Root of x squared minus a squared",
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Reciprocal of Square of Hyperbolic Cosine of a x"
] |
proofwiki-9343 | Primitive of Reciprocal of x cubed by Root of x squared minus a squared | :$\ds \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {z - a^2} }
| c = In... | :$\ds \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} } = \frac {\sqrt {x^2 - a^2} } {2 a^2 x^2} + \frac 1 {2 a^3} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {x^2 - a^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {z - a^2} }
| c ... | Primitive of Reciprocal of x cubed by Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9344 | Primitive of x by Root of x squared minus a squared | :$\ds \int x \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {x^2 - a^2} \rd x
| r = \int \frac {\sqrt z \sqrt {z - a^2} \rd z} {2 \sqrt z}
| c = Integration by... | :$\ds \int x \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {x^2 - a^2} \rd x
| r = \int \frac {\sqrt z \sqrt {z - a^2} \rd z} {2 \sqrt z}
| c = [[Integrat... | Primitive of x by Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b"
] |
proofwiki-9345 | Primitive of x squared by Root of x squared minus a squared | :$\ds \int x^2 \sqrt {x^2 - a^2} \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 - \frac {a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x \ge a$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
:$x \ge a$
or:
:$x \le -a$
where it is assumed that $a > 0$.
First let $x \ge a$.
Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Deri... | :$\ds \int x^2 \sqrt {x^2 - a^2} \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 + \frac {a^2 x \sqrt {x^2 - a^2} } 8 - \frac {a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $\size x \ge a$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 \ge a^2$, that is, either:
:$x \ge a$
or:
:$x \le -a$
where it is assumed that $a > 0$.
First let $x \ge a$.
Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule fo... | Primitive of x squared by Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_minus_a_squared | [
"Primitive of x squared by Root of x squared minus a squared",
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b by Root of p x + q",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q",
"Integration by Substitution",
"Negative of Logarithm of x plus Root x squared mi... |
proofwiki-9346 | Primitive of x cubed by Root of x squared minus a squared | :$\ds \int x^3 \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {x^2 - a^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {z - a^2} \rd z} {2 \sqrt z}
| c = Integration ... | :$\ds \int x^3 \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {x^2 - a^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {z - a^2} \rd z} {2 \sqrt z}
| c = [[Integr... | Primitive of x cubed by Root of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_minus_a_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of x by Root of a x + b"
] |
proofwiki-9347 | Primitive of Root of x squared minus a squared over x | :$\ds \int \frac {\sqrt {x^2 - a^2} } x \rd x = \sqrt {x^2 - a^2} - a \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } x \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 \sqrt z \sqrt z}
| c = Integ... | :$\ds \int \frac {\sqrt {x^2 - a^2} } x \rd x = \sqrt {x^2 - a^2} - a \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } x \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 \sqrt z \sqrt z}
| c = [... | Primitive of Root of x squared minus a squared over x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9348 | Primitive of Root of x squared minus a squared over x cubed | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 z^{3/2} \sqrt z}
| c = I... | :$\ds \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x = \frac {-\sqrt {x^2 - a^2} } {2 x^2} + \frac 1 {2 a} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {x^2 - a^2} } {x^3} \rd x
| r = \int \frac {\sqrt {z - a^2} \rd z} {2 z^{3/2} \sqrt z}
| c... | Primitive of Root of x squared minus a squared over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_over_x_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9349 | Primitive of Reciprocal of Root of x squared minus a squared cubed | :$\ds \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \paren {\sqrt {z - a^2} }^3}
... | :$\ds \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {a^2 \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \paren {\sqrt {z - a^2} }^3}
... | Primitive of Reciprocal of Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q"
] |
proofwiki-9350 | Primitive of x over Root of x squared minus a squared cubed | :$\ds \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {\sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {x \rd z} {2 x z^{3/2} }
| c = Integ... | :$\ds \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {\sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {x \rd z} {2 x z^{3/2} }
| c = [... | Primitive of x over Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9351 | Primitive of x squared over Root of x squared minus a squared cubed | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-... | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \frac {-x} {\sqrt {x^2 - a^2} } + \ln \size {x + \sqrt {x^2 - a^2} } + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
and let:
{... | Primitive of x squared over Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of x squared minus a squared cubed",
"Integration by Parts",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form"
] |
proofwiki-9352 | Primitive of x cubed over Root of x squared minus a squared cubed | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 -... | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 -... | Primitive of x cubed over Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Root of x squared minus a squared",
"Primitive of x over Root of x squared minus a squared cubed"
] |
proofwiki-9353 | Primitive of Reciprocal of x by Root of x squared minus a squared cubed | :$\ds \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {z - a^2}... | :$\ds \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3} = \frac {-1} {a^2 \sqrt {x^2 - a^2} } - \frac 1 {a^3} \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {z - ... | Primitive of Reciprocal of x by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x by Power of Root of a x + b",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9354 | Primitive of Reciprocal of x squared by Root of x squared minus a squared cubed | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3} = \frac {-\sqrt {x^2 - a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3} = \frac {-\sqrt {x^2 - a^2} } {a^4 x} - \frac x {a^4 \sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^2 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x squared by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by Root of x squared minus a squared",
"Primitive of Reciprocal of Root of x squared minus a squared cubed"
] |
proofwiki-9355 | Primitive of Reciprocal of x cubed by Root of x squared minus a squared cubed | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3} = \frac 1 {2 a^2 x^2 \sqrt {x^2 - a^2} } - \frac 3 {2 a^4 \sqrt {x^2 - a^2} } + \frac 3 {2 a^5} \arcsec \size {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3} = \frac 1 {2 a^2 x^2 \sqrt {x^2 - a^2} } - \frac 3 {2 a^4 \sqrt {x^2 - a^2} } + \frac 3 {2 a^5} \arcsec \size {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {x^2 - a^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x^3 \paren {\sqrt {x^2 - a^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x cubed by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x cubed by Root of x squared minus a squared",
"Primitive of Reciprocal of x by Root of x squared minus a squared cubed"
] |
proofwiki-9356 | Primitive of Root of x squared minus a squared cubed | :$\ds \int \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 \sqrt z} \rd x
| c = ... | :$\ds \int \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 \sqrt z} \rd x
| ... | Primitive of Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q over Root of a x + b",
"Primitive of Root of p x + q over Root of a x + b",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form"
] |
proofwiki-9357 | Primitive of x by Root of x squared minus a squared cubed | :$\ds \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {z^{3/2} } 2 \rd z
| c = Integration by Substi... | :$\ds \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {z^{3/2} } 2 \rd z
| c = [[Integration by ... | Primitive of x by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9358 | Primitive of x squared by Root of x squared minus a squared cubed | :$\ds \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {z - a^2}^{3/2} } 2 \rd z
| c = In... | :$\ds \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {x^2 - a^2} }^3} {24} - \frac {a^4 x \sqrt {x^2 - a^2} } {16} + \frac {a^6} {16} \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {z - a^2}^{3/2} } 2 \rd z
| c ... | Primitive of x squared by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q by Root of a x + b",
"Primitive of Root of x squared minus a squared cubed"
] |
proofwiki-9359 | Primitive of x cubed by Root of x squared minus a squared cubed | :$\ds \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x
| c = Primitive of Power
}}
{{eqn | r = \int x \paren {x^2 - a^2} \paren {\sqrt... | :$\ds \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {x^2 - a^2} }^3 \rd x
| c = [[Primitive of Power]]
}}
{{eqn | r = \int x \paren {x^2 - a^2} \paren {\... | Primitive of x cubed by Root of x squared minus a squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_minus_a_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_x_squared_minus_a_squared_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Primitive of Power",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9360 | Primitive of Root of x squared minus a squared cubed over x | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 - a^2 \sqrt {x^2 - a^2} + a^3 \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z} \rd z
| ... | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} x \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 - a^2 \sqrt {x^2 - a^2} + a^3 \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z} \rd z
... | Primitive of Root of x squared minus a squared cubed over x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of Root of a x + b over x",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] |
proofwiki-9361 | Primitive of Root of x squared minus a squared cubed over x squared | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x + \frac{3 x \sqrt {x^2 - a^2} } 2 - \frac {3 a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z \sqrt z} \r... | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} x + \frac{3 x \sqrt {x^2 - a^2} } 2 - \frac {3 a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z \sqrt z... | Primitive of Root of x squared minus a squared cubed over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x_squared | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of x squared minus a squared"
] |
proofwiki-9362 | Primitive of Root of x squared minus a squared cubed over x cubed | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 - a^2} } 2 - \frac {3 a} 2 \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z^2} \rd z
... | :$\ds \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {x^2 - a^2} }^3} {2 x^2} + \frac {3 \sqrt {x^2 - a^2} } 2 - \frac {3 a} 2 \arcsec \size {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {x^2 - a^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {z - a^2} }^3} {2 z^2} \rd ... | Primitive of Root of x squared minus a squared cubed over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_x_squared_minus_a_squared_cubed_over_x_cubed | [
"Primitives involving Root of x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of x squared minus a squared over x"
] |
proofwiki-9363 | Primitive of x over Root of a squared minus x squared | :$\ds \int \frac {x \rd x} {\sqrt {a^2 - x^2} } = -\sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = -2 x
| c = Chain Rule for Derivatives, Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {a^2 - x^2} }
| r = \int -\frac {z \... | :$\ds \int \frac {x \rd x} {\sqrt {a^2 - x^2} } = -\sqrt {a^2 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z^2
| r = a^2 - x^2
| c =
}}
{{eqn | ll= \leadsto
| l = 2 z \frac {\d z} {\d x}
| r = -2 x
| c = [[Chain Rule for Derivatives]], [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\sqrt {a^2 - x^2} }
| r = \int -\f... | Primitive of x over Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant"
] |
proofwiki-9364 | Primitive of x cubed over Root of a squared minus x squared | :$\ds \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 - a^2 \sqrt {a^2 - x^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{end-eqn}}
... | :$\ds \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 - a^2 \sqrt {a^2 - x^2} + C$ | 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 = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{end-e... | Primitive of x cubed over Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of a squared minus x squared",
"Integration by Parts",
"Primitive of x by Root of a squared minus x squared"
] |
proofwiki-9365 | Primitive of Reciprocal of x squared by Root of a squared minus x squared | :$\ds \int \frac {\d x} {x^2 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {a^2 x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^2 \sqrt {a^2 - x^2} }
| r = \int \frac {\d z} {2 z \sqrt z \sqrt {a^2 - z} }
| c = Integrat... | :$\ds \int \frac {\d x} {x^2 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {a^2 x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^2 \sqrt {a^2 - x^2} }
| r = \int \frac {\d z} {2 z \sqrt z \sqrt {a^2 - z} }
| c = [[In... | Primitive of Reciprocal of x squared by Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of Power of x by Root of a x + b"
] |
proofwiki-9366 | Primitive of Reciprocal of x cubed by Root of a squared minus x squared | :$\ds \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {a^2 - z} }
| c = In... | :$\ds \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} } = \frac {-\sqrt {a^2 - x^2} } {2 a^2 x^2} - \frac 1 {2 a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x^3 \sqrt {a^2 - x^2} }
| r = \int \frac {\d z} {2 z^{3/2} \sqrt z \sqrt {a^2 - z} }
| c ... | Primitive of Reciprocal of x cubed by Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of Power of x by Root of a x + b",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form"
] |
proofwiki-9367 | Primitive of Root of a squared minus x squared/Arcsine Form | :$\ds \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cos \theta
| c = Derivative of Sine Function
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | ll= \leadsto
| l = a^2 - x^2... | :$\ds \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | n = 1
| ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \cos \theta
| c = [[Derivative of Sine Function]]
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin \theta
}}
{{eqn | ll= \leadsto
| l = a^2... | Primitive of Root of a squared minus x squared/Arcsine Form | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared/Arcsine_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared/Arcsine_Form | [
"Primitive of Root of a squared minus x squared"
] | [] | [
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9368 | Primitive of x by Root of a squared minus x squared | :$\ds \int x \sqrt {a^2 - x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {a^2 - x^2} \rd x
| r = \int \frac {\sqrt z \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = Integration by... | :$\ds \int x \sqrt {a^2 - x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \sqrt {a^2 - x^2} \rd x
| r = \int \frac {\sqrt z \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = [[Integrat... | Primitive of x by Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b"
] |
proofwiki-9369 | Primitive of x squared by Root of a squared minus x squared | :$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$ | Let us assume that $a > 0$.
Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d t}
| r = a \cos t
| c =
}}
{{eqn | ll= \leadsto
| l = a^2 - x^2
| r = a^2 - a^2 \sin^2 t
| c =
}}
{{eqn | r = a^2 \paren {1 - \sin^2 t}
... | :$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$ | Let us assume that $a > 0$.
Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d t}
| r = a \cos t
| c =
}}
{{eqn | ll= \leadsto
| l = a^2 - x^2
| r = a^2 - a^2 \sin^2 t
| c =
}}
{{eqn | r = a^2 \paren {1 - \sin^2 t}
... | Primitive of x squared by Root of a squared minus x squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared/Proof_1 | [
"Primitive of x squared by Root of a squared minus x squared",
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Integration by Substitution",
"Primitive of Sine of a x squared by Cosine of a x squared",
"Quadruple Angle Formulas/Sine",
"Distributive Laws/Arithmetic"
] |
proofwiki-9370 | Primitive of x squared by Root of a squared minus x squared | :$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^2 \sqrt {a^2 - x^2} \rd x
| r = \int \frac {z \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = Integration by Substitu... | :$\ds \int x^2 \sqrt {a^2 - x^2} \rd x = -\frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \arcsin \frac x a + C$ | {{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^2 \sqrt {a^2 - x^2} \rd x
| r = \int \frac {z \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = [[Integration by Su... | Primitive of x squared by Root of a squared minus x squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared/Proof_2 | [
"Primitive of x squared by Root of a squared minus x squared",
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b by Root of p x + q",
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/a greater than 0, p less than 0"
] |
proofwiki-9371 | Primitive of x cubed by Root of a squared minus x squared | :$\ds \int x^3 \sqrt {a^2 - x^2} \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^5} 5 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {a^2 - x^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = Integration ... | :$\ds \int x^3 \sqrt {a^2 - x^2} \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^5} 5 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^3} 3 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^3 \sqrt {a^2 - x^2} \rd x
| r = \int \frac {z^{3/2} \sqrt {a^2 - z} \rd z} {2 \sqrt z}
| c = [[Integr... | Primitive of x cubed by Root of a squared minus x squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_a_squared_minus_x_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_a_squared_minus_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of x by Root of a x + b"
] |
proofwiki-9372 | Primitive of Root of a squared minus x squared over x | :$\ds \int \frac {\sqrt {a^2 - x^2} } x \rd x = \sqrt {a^2 - x^2} - a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } x \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 \sqrt z \sqrt z}
| c = Integ... | :$\ds \int \frac {\sqrt {a^2 - x^2} } x \rd x = \sqrt {a^2 - x^2} - a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } x \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 \sqrt z \sqrt z}
| c = [... | Primitive of Root of a squared minus x squared over x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x | [
"Primitive of Root of a squared minus x squared over x",
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form"
] |
proofwiki-9373 | Primitive of Root of a squared minus x squared over x squared | :$\ds \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x = \frac {-\sqrt {a^2 - x^2} } x - \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 z \sqrt z}
| c = Integra... | :$\ds \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x = \frac {-\sqrt {a^2 - x^2} } x - \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } {x^2} \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 z \sqrt z}
| c = [[I... | Primitive of Root of a squared minus x squared over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-9374 | Primitive of Root of a squared minus x squared over x cubed | :$\ds \int \frac {\sqrt {a^2 - x^2} } {x^3} \rd x = \frac {-\sqrt {a^2 - x^2} } {2 x^2} + \frac 1 {2 a} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } {x^3} \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 z^{3/2} \sqrt z}
| c = I... | :$\ds \int \frac {\sqrt {a^2 - x^2} } {x^3} \rd x = \frac {-\sqrt {a^2 - x^2} } {2 x^2} + \frac 1 {2 a} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\sqrt {a^2 - x^2} } {x^3} \rd x
| r = \int \frac {\sqrt {a^2 - z} \rd z} {2 z^{3/2} \sqrt z}
| c... | Primitive of Root of a squared minus x squared over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_over_x_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Root of a x + b over Power of x/Formulation 1",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form"
] |
proofwiki-9375 | Primitive of x over Root of a squared minus x squared cubed | :$\ds \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {x \rd z} {-2 x z^{3/2} }
| c = Int... | :$\ds \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {\sqrt {a^2 - x^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {x \rd z} {-2 x z^{3/2} }
| c =... | Primitive of x over Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9376 | Primitive of x squared over Root of a squared minus x squared cubed | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-... | :$\ds \int \frac {x^2 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \frac x {\sqrt {a^2 - x^2} } - \arcsin \frac x a + C$ | 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 = x
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
and let:
{... | Primitive of x squared over Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_over_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Primitive of x over Root of a squared minus x squared cubed",
"Integration by Parts",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-9377 | Primitive of x cubed over Root of a squared minus x squared cubed | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \sqrt {a^2 - x^2} + \frac {a^2} {\sqrt {a^2 - x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = -\int \frac {x \paren {a^2 ... | :$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \sqrt {a^2 - x^2} + \frac {a^2} {\sqrt {a^2 - x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {x \paren {x^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2 + a^2} \rd x} {\paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = -\int \frac {x \paren {a^2 ... | Primitive of x cubed over Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over Root of a squared minus x squared",
"Primitive of x over Root of a squared minus x squared cubed"
] |
proofwiki-9378 | Primitive of Reciprocal of x by Root of a squared minus x squared cubed | :$\ds \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {a^2 - z}... | :$\ds \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3} = \frac 1 {a^2 \sqrt {a^2 - x^2} } - \frac 1 {a^3} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {x \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {\d z} {2 \sqrt z \sqrt z \paren {\sqrt {a^2 ... | Primitive of Reciprocal of x by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x by Power of Root of a x + b",
"Primitive of Reciprocal of x by Root of a squared minus x squared"
] |
proofwiki-9379 | Primitive of Reciprocal of x squared by Root of a squared minus x squared cubed | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {a^2 - x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2 + x^2} \rd x} {a^2 x^2 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^2 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-\sqrt {a^2 - x^2} } {a^4 x} + \frac x {a^4 \sqrt {a^2 - x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^2 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2 + x^2} \rd x} {a^2 x^2 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x squared by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared by Root of a squared minus x squared",
"Primitive of Reciprocal of Root of a squared minus x squared cubed"
] |
proofwiki-9380 | Primitive of Reciprocal of x cubed by Root of a squared minus x squared cubed | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2 + x^2} \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | :$\ds \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3} = \frac {-1} {2 a^2 x^2 \sqrt {a^2 - x^2} } + \frac 3 {2 a^4 \sqrt {a^2 - x^2} } - \frac 3 {2 a^5} \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {\sqrt {a^2 - x^2} }^3}
| r = \int \frac {a^2 \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \int \frac {\paren {a^2 - x^2 + x^2} \rd x} {a^2 x^3 \paren {\sqrt {a^2 - x^2} }^3}
| c =
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x cubed by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x cubed by Root of a squared minus x squared",
"Primitive of Reciprocal of x by Root of a squared minus x squared cubed"
] |
proofwiki-9381 | Primitive of Root of a squared minus x squared cubed | :$\ds \int \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2 x \sqrt {a^2 - x^2} } 8 + \frac {3 a^4} 8 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 \sqrt z} \rd x
| c = ... | :$\ds \int \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {x \paren {\sqrt {a^2 - x^2} }^3} 4 + \frac {3 a^2 x \sqrt {a^2 - x^2} } 8 + \frac {3 a^4} 8 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 \sqrt z} \rd x
| ... | Primitive of Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q over Root of a x + b",
"Primitive of Root of p x + q over Root of a x + b",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-9382 | Primitive of x by Root of a squared minus x squared cubed | :$\ds \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {z^{3/2} } {-2} \rd z
| c = Integration by Su... | :$\ds \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^5} 5 + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = a^2 - x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = -2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {z^{3/2} } {-2} \rd z
| c = [[Integration... | Primitive of x by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9383 | Primitive of x squared by Root of a squared minus x squared cubed | :$\ds \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {a^2 - z}^{3/2} } 2 \rd z
| c = In... | :$\ds \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int \frac {\sqrt z \paren {a^2 - z}^{3/2} } 2 \rd z
| c ... | Primitive of x squared by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of p x + q by Root of a x + b",
"Primitive of Root of a squared minus x squared cubed"
] |
proofwiki-9384 | Primitive of x cubed by Root of a squared minus x squared cubed | :$\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x
| c = Primitive of Power
}}
{{eqn | r = -\int x \paren {a^2 - x^2} \paren {\sqr... | :$\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$ | {{begin-eqn}}
{{eqn | l = \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x
| r = \int x \paren {x^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x
| c =
}}
{{eqn | r = \int x \paren {x^2 - a^2 + a^2} \paren {\sqrt {a^2 - x^2} }^3 \rd x
| c = [[Primitive of Power]]
}}
{{eqn | r = -\int x \paren {a^2 - x^2} \paren {... | Primitive of x cubed by Root of a squared minus x squared cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_a_squared_minus_x_squared_cubed | https://proofwiki.org/wiki/Primitive_of_x_cubed_by_Root_of_a_squared_minus_x_squared_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Primitive of Power",
"Linear Combination of Integrals/Indefinite",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9385 | Primitive of Root of a squared minus x squared cubed over x | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} x \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 + a^2 \sqrt {a^2 - x^2} - a^3 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 z} \rd z
| ... | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} x \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 + a^2 \sqrt {a^2 - x^2} - a^3 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} x \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 z} \rd z
... | Primitive of Root of a squared minus x squared cubed over x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of Root of a x + b over x",
"Primitive of Root of a x + b over x",
"Primitive of Reciprocal of x by Root of a squared minus x squared/Logarithm Form"
] |
proofwiki-9386 | Primitive of Root of a squared minus x squared cubed over x squared | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} x - \frac {3 x \sqrt {a^2 - x^2} } 2 - \frac {3 a^2} 2 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 z \sqrt z} \r... | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} x - \frac {3 x \sqrt {a^2 - x^2} } 2 - \frac {3 a^2} 2 \arcsin \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^2} \rd x
| r = \int \frac {\paren {\sqrt {a^2 - z} }^3} {2 z \sqrt z... | Primitive of Root of a squared minus x squared cubed over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x_squared | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of a squared minus x squared"
] |
proofwiki-9387 | Primitive of Root of a squared minus x squared cubed over x cubed | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} {2 x^2} - \frac {3 \sqrt {a^2 - x^2} } 2 + \frac {3 a} 2 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {2 z^2} \rd z
... | :$\ds \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^3} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} {2 x^2} - \frac {3 \sqrt {a^2 - x^2} } 2 + \frac {3 a} 2 \map \ln {\frac {a + \sqrt {a^2 - x^2} } {\size x} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {x^3} \rd x
| r = \int \frac {\paren {\sqrt {a^2 - x^2} }^3} {2 z^2} \r... | Primitive of Root of a squared minus x squared cubed over x cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x_cubed | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared_cubed_over_x_cubed | [
"Primitives involving Root of a squared minus x squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power of a x + b over Power of p x + q/Formulation 3",
"Primitive of Root of a squared minus x squared over x"
] |
proofwiki-9388 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | First:
{{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}
| c = Completing the Square
}}
{{eqn | n = 1
| ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {4 a \rd x} {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} }
| ... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | First:
{{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}
| c = [[Completing the Square]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {4 a \rd x} {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} }
... | Primitive of Reciprocal of a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Completing the Square",
"Power Rule for Derivatives",
"Integration by Substitution"
] |
proofwiki-9389 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | We aim to use Primitive of $\dfrac 1 {a x^2 + b x + c}$ with:
{{begin-eqn}}
{{eqn | l = a
| r = 3
}}
{{eqn | l = b
| r = 4
}}
{{eqn | l = c
| r = 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = b^2 - 4 a c
| r = 4^2 - 4 \times 3 \times 2
}}
{{eqn | r = 16 - 24
}}
{{eqn | r = -8
}}
{{end... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | We aim to use [[Primitive of Reciprocal of a x squared plus b x plus c|Primitive of $\dfrac 1 {a x^2 + b x + c}$]] with:
{{begin-eqn}}
{{eqn | l = a
| r = 3
}}
{{eqn | l = b
| r = 4
}}
{{eqn | l = c
| r = 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = b^2 - 4 a c
| r = 4^2 - 4 \times... | Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Examples/3_x^2_+_4_x_+_2/Proof_1 | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9390 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {3 x^2 + 4 x + 2}
| r = \dfrac 1 3 \int \frac {\d x} {x^2 + \frac 4 3 x + \frac 2 3}
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac 2 3 - \frac 4 9} }
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {3 x^2 + 4 x + 2}
| r = \dfrac 1 3 \int \frac {\d x} {x^2 + \frac 4 3 x + \frac 2 3}
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac 2 3 - \frac 4 9} }
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x... | Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Examples/3_x^2_+_4_x_+_2/Proof_2 | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9391 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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 | 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",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"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-9392 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | Let $b = 0$.
From Primitive of Reciprocal of a x squared plus b x plus 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 ... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | Let $b = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus 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 {\d... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_2 | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9393 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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
... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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 | 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",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b"
] |
proofwiki-9394 | Primitive of Reciprocal of a x squared plus b x plus c | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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 - ... | :<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 - \sqrt {b^2 - 4 a c} } {2 a x + b + \sqrt {b^2 - 4 a c} } } + C & : b^... | 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 | 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",
"Primitives involving a x squared plus b x plus c",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-9395 | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0 | Let $b = 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \\
\dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \\
\dfrac {-1} {a x} + C & : c = 0
\end {... | 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 $b = 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \\
\dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \\
\dfrac {-1} {a x} + C & : c = 0
\end ... | 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 | 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-9396 | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0 | Let $b = 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \\
\dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \\
\dfrac {-1} {a x} + C & : c = 0
\end {... | Let $b = 0$.
From Primitive of Reciprocal of a x squared plus b x plus 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 ... | Let $b = 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 1 {\sqrt {a c} } \map \arctan {x \sqrt {\dfrac a c} } + C & : a c > 0 \\ \\
\dfrac 1 {2 \sqrt {-a c} } \ln \size {\dfrac {a x - \sqrt {-a c} } {a x + \sqrt {-a c} } } + C & : a c < 0 \\ \\
\dfrac {-1} {a x} + C & : c = 0
\end ... | Let $b = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus c]], we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\d... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_2 | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-9397 | Primitive of Reciprocal of a x squared plus b x plus c/a equal to 0 | :$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {b x + c} + C$
when $a = 0$. | {{begin-eqn}}
{{eqn | l = a
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {b x + c}
| c =
}}
{{eqn | r = \frac 1 b \ln \size {b x + c} + C
| c = Primitive of $\dfrac 1 {a x + b}$
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 1 b \ln \size {b x + c} + C$
when $a = 0$. | {{begin-eqn}}
{{eqn | l = a
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {b x + c}
| c =
}}
{{eqn | r = \frac 1 b \ln \size {b x + c} + C
| c = [[Primitive of Reciprocal of a x + b|Primitive of $\dfrac 1 {a x + b}$]]
}}
{{e... | Primitive of Reciprocal of a x squared plus b x plus c/a equal to 0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/a_equal_to_0 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/a_equal_to_0 | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] | [] | [
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9398 | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2 | Let $a \in \R_{\ne 0}$.
{{:Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0}} | Let $b = 0$.
From Primitive of Reciprocal of a x squared plus b x plus 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 ... | Let $a \in \R_{\ne 0}$.
{{:Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0}} | Let $b = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus 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 {\d... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_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"
] |
proofwiki-9399 | Primitive of x over a x squared plus b x plus c | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {a x^2 + b x + c} = \frac 1 {2 a} \ln \size {a x^2 + b x + c} - \frac b {2 a} \int \frac {\d x} {a x^2 + b x + c}$ | First note that by Derivative of Power:
:$(1): \quad \map {\dfrac \d {\d x} } {a x^2 + b x + c} = 2 a x + b$
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x^2 + b x + c}
| r = \frac 1 {2 a} \int \frac {2 a x \rd x} {a x^2 + b x + c}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \... | Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {a x^2 + b x + c} = \frac 1 {2 a} \ln \size {a x^2 + b x + c} - \frac b {2 a} \int \frac {\d x} {a x^2 + b x + c}$ | First note that by [[Derivative of Power]]:
:$(1): \quad \map {\dfrac \d {\d x} } {a x^2 + b x + c} = 2 a x + b$
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {a x^2 + b x + c}
| r = \frac 1 {2 a} \int \frac {2 a x \rd x} {a x^2 + b x + c}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{e... | Primitive of x over a x squared plus b x plus c | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_squared_plus_b_x_plus_c | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_squared_plus_b_x_plus_c | [
"Primitive of x over 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",
"Primitive of Function under its Derivative"
] |
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