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-9200 | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form | :$\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \tanh^{-1} \frac x a + C$
where $\size x < a$. | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | :$\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \tanh^{-1} \frac x a + C$
where $\size x < a$. | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form/Proof | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Integral of Constant"
] |
proofwiki-9201 | Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \arsinh {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh {\frac x a}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sinh u
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = a \cosh u
| c = Derivative of Hyperbolic Sine
}}
{{eqn | ll=... | :$\ds \int \frac {\d x} {\sqrt {x^2 + a^2} } = \arsinh {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arsinh {\frac x a}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \sinh u
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = a \cosh u
| c = [[Derivative of Hyperbolic Sine]]
}}
{{eqn |... | Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | [
"Primitive of Reciprocal of Root of x squared plus a squared"
] | [] | [
"Derivative of Hyperbolic Sine",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integral of Constant"
] |
proofwiki-9202 | Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \arcosh {\size {\frac x a} } + C$
for $x^2 > a^2$. | When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and then $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.
When $\size x < a$ we have that $x^2 - a^2 < 0$ and then $\sqrt {x^2 - a^2}$ is not defined.
Hence the domain needs to be restricted to $\size x > a$, or that is: $\size {\dfrac x a} > 1$.
Let $x > a$.
Let:
{{begi... | :$\ds \int \frac {\d x} {\sqrt {x^2 - a^2} } = \dfrac {\size x} x \arcosh {\size {\frac x a} } + C$
for $x^2 > a^2$. | When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and then $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.
When $\size x < a$ we have that $x^2 - a^2 < 0$ and then $\sqrt {x^2 - a^2}$ is not defined.
Hence the [[Definition:Domain of Real Function|domain]] needs to be restricted to $\size x > a$, or that is: $\size {\... | Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cosine_Form | [
"Expressions whose Primitives are Inverse Hyperbolic Functions",
"Primitives involving Inverse Hyperbolic Cosine Function",
"Primitive of Reciprocal of Root of x squared minus a squared"
] | [] | [
"Definition:Real Function/Domain",
"Derivative of Hyperbolic Cosine Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integral of Constant",
"Integration by Substitution",
"Primitive of Constant Multiple of F... |
proofwiki-9203 | Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \csch^{-1} {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \csch^{-1} {\frac x a}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \csch u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = -a \csch u \coth u
| c = Derivative of Hyperbolic Cosecant
}}
{{eqn | ll = \leadsto
| ... | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \csch^{-1} {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \csch^{-1} {\frac x a}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = a \csch u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}
| r = -a \csch u \coth u
| c = [[Derivative of Hyperbolic Cosecant]]
}}
{{eqn | ll = \leadsto
... | Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Cosecant_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Cosecant_Form | [
"Primitive of Reciprocal of x by Root of x squared plus a squared",
"Expressions whose Primitives are Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cosecant"
] | [] | [
"Derivative of Hyperbolic Cosecant",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-9204 | Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form | For $a > 0$ and $0 < \size x < a$:
:$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C$ | We note that $\sech^{-1} \dfrac x a$ is defined for $x \in \hointl 0 a$.
Hence we treat the two cases $x > 0$ and $x < 0$ separately.
First let $x > 0$.
Note that $\dfrac 1 {x \sqrt {a^2 - x^2} }$ is not defined at $\pm a$, so we are concerned only about the interval $\openint 0 a$.
Then:
{{begin-eqn}}
{{eqn | l = u
... | For $a > 0$ and $0 < \size x < a$:
:$\ds \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \sech^{-1} {\frac {\size x} a} + C$ | We note that $\sech^{-1} \dfrac x a$ is defined for $x \in \hointl 0 a$.
Hence we treat the two cases $x > 0$ and $x < 0$ separately.
First let $x > 0$.
Note that $\dfrac 1 {x \sqrt {a^2 - x^2} }$ is not defined at $\pm a$, so we are concerned only about the [[Definition:Open Real Interval|interval]] $\openint 0 a$... | Primitive of Reciprocal of x by Root of a squared minus x squared/Inverse Hyperbolic Secant Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Secant_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Secant_Form | [
"Expressions whose Primitives are Inverse Hyperbolic Functions",
"Inverse Hyperbolic Secant",
"Primitive of Reciprocal of x by Root of a squared minus x squared"
] | [] | [
"Definition:Real Interval/Open",
"Derivative of Hyperbolic Secant",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Integral of Constant",
"Derivative of Constant Multiple",
"Integration by Substitution",
"Primitive of ... |
proofwiki-9205 | Primitive of x by Root of a x + b | :$\ds \int x \sqrt {a x + b} \rd x = \frac {2 \paren {3 a x - 2 b} } {15 a^2} \sqrt {\paren {a x + b}^3}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sqrt {a x + b} \rd x
| r = \frac 2 a \int \frac {u^2 - b} a u^2 \rd u
| c = Primitive of Function ... | :$\ds \int x \sqrt {a x + b} \rd x = \frac {2 \paren {3 a x - 2 b} } {15 a^2} \sqrt {\paren {a x + b}^3}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sqrt {a x + b} \rd x
| r = \frac 2 a \int \frac {u^2 - b} a u^2 \rd u
| c = [[Primitive of Funct... | Primitive of x by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Definition:Common Denominator"
] |
proofwiki-9206 | Laplace Transform of Heaviside Step Function times Function | Let $\map f t: \R \to \R$ or $\R \to \C$ be a function of exponential order $a$ for some constant $a \in \R$.
Let $f$ be piecewise continuous with one-sided limits on any closed interval of the form $\closedint 0 b$ where $b > 0$.
Let $\map {u_c} t$ be the Heaviside step function.
Let $\laptrans {\map f t} = \map F s$ ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map {u_c} t \, \map f {t - c} }
| r = \int_0^{\to +\infty} \map {u_c} t e^{-s t} \map f {t - c} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to c^-} \map {u_c} t e^{-s t} \map f {t - c} \rd t + \int_{\to c^+}^{\to +\infty} \map {u_c} t e^{-s t} \map f... | Let $\map f t: \R \to \R$ or $\R \to \C$ be a [[Definition:Function|function]] of [[Definition:Exponential Order|exponential order $a$]] for some [[Definition:Constant|constant]] $a \in \R$.
Let $f$ be [[Definition:Piecewise Continuous Function with One-Sided Limits|piecewise continuous with one-sided limits]] on any ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map {u_c} t \, \map f {t - c} }
| r = \int_0^{\to +\infty} \map {u_c} t e^{-s t} \map f {t - c} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to c^-} \map {u_c} t e^{-s t} \map f {t - c} \rd t + \int_{\to c^+}^{\to +\infty} \map {u_c} t e^{-s t} \map f... | Laplace Transform of Heaviside Step Function times Function | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function_times_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function_times_Function | [
"Heaviside Step Function",
"Examples of Laplace Transforms"
] | [
"Definition:Function",
"Definition:Exponential Order",
"Definition:Constant",
"Definition:Piecewise Continuous Function/One-Sided Limits",
"Definition:Real Interval/Closed",
"Definition:Heaviside Step Function",
"Definition:Laplace Transform"
] | [
"Sum of Complex Integrals on Adjacent Intervals",
"Integration by Substitution",
"Definite Integral on Zero Interval",
"Sum of Complex Integrals on Adjacent Intervals",
"Exponent Combination Laws"
] |
proofwiki-9207 | Primitive of x squared by Root of a x + b | :$\ds \int x^2 \sqrt {a x + b} \rd x = \frac {2 \paren {15 a^2 x^2 - 12 a b x + 8 b^2} } {105 a^3} \sqrt {\paren {a x + b}^3} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sqrt {a x + b} \rd x
| r = \frac 2 a \int \paren {\frac {u^2 - b} a}^2 u^2 \rd u
| c = Primitive o... | :$\ds \int x^2 \sqrt {a x + b} \rd x = \frac {2 \paren {15 a^2 x^2 - 12 a b x + 8 b^2} } {105 a^3} \sqrt {\paren {a x + b}^3} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sqrt {a x + b} \rd x
| r = \frac 2 a \int \paren {\frac {u^2 - b} a}^2 u^2 \rd u
| c = [[Primiti... | Primitive of x squared by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Definition:Common Denominator"
] |
proofwiki-9208 | Primitive of Root of a x + b over x | :$\ds \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Putting $m = -1$ and $n = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \int \frac... | :$\ds \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd... | Primitive of Root of a x + b over x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x/Proof_1 | [
"Primitive of Root of a x + b over x",
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b"
] |
proofwiki-9209 | Primitive of Root of a x + b over x | :$\ds \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }$ | Let:
{{begin-eqn}}
{{eqn | l = v
| r = \sqrt x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = \frac 1 {2 \sqrt x}
| c = Power Rule for Derivatives
}}
{{eqn | l = u
| r = \frac {2 \sqrt {a x + b} } {\sqrt x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u}... | :$\ds \int \frac {\sqrt {a x + b} } x \rd x = 2 \sqrt {a x + b} + b \int \frac {\d x} {x \sqrt {a x + b} }$ | Let:
{{begin-eqn}}
{{eqn | l = v
| r = \sqrt x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = \frac 1 {2 \sqrt x}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = u
| r = \frac {2 \sqrt {a x + b} } {\sqrt x}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\... | Primitive of Root of a x + b over x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x/Proof_2 | [
"Primitive of Root of a x + b over x",
"Primitives involving Root of a x + b"
] | [] | [
"Power Rule for Derivatives",
"Quotient Rule for Derivatives",
"Integration by Parts"
] |
proofwiki-9210 | Primitive of Root of a x + b over x squared | :$\ds \int \frac {\sqrt {a x + b} } {x^2} \rd x = -\frac {\sqrt {a x + b} } x + \frac a 2 \int \frac {\d x} {x \sqrt {a x + b} }$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Putting $m = -2$ and $n = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \int \frac... | :$\ds \int \frac {\sqrt {a x + b} } {x^2} \rd x = -\frac {\sqrt {a x + b} } x + \frac a 2 \int \frac {\d x} {x \sqrt {a x + b} }$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd... | Primitive of Root of a x + b over x squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_x_squared | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b",
"Primitive of Reciprocal of x squared by Root of a x + b"
] |
proofwiki-9211 | Primitive of Power of x over Root of a x + b | :$\ds \int \frac {x^m} {\sqrt {a x + b} } \rd x = \frac {2 x^m \sqrt{a x + b} } {\paren {2 m + 1} a} - \frac {2 m b} {\paren {2 m + 1} a} \int \frac {x^{m - 1} } {\sqrt {a x + b} } \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Putting $n = -\dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \int ... | :$\ds \int \frac {x^m} {\sqrt {a x + b} } \rd x = \frac {2 x^m \sqrt{a x + b} } {\paren {2 m + 1} a} - \frac {2 m b} {\paren {2 m + 1} a} \int \frac {x^{m - 1} } {\sqrt {a x + b} } \rd x$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x... | Primitive of Power of x over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9212 | Primitive of Reciprocal of Power of x by Root of a x + b | :$\ds \int \frac {\d x} {x^m \sqrt {a x + b} } = -\frac {\sqrt {a x + b} } {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 3} a} {\paren {2 m - 2} b} \int \frac {\d x} {x^{m - 1} \sqrt {a x + b} }$ | From Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$
Putting $n := -\dfrac 1 2$ and $m := -m$:
{... | :$\ds \int \frac {\d x} {x^m \sqrt {a x + b} } = -\frac {\sqrt {a x + b} } {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 3} a} {\paren {2 m - 2} b} \int \frac {\d x} {x^{m - 1} \sqrt {a x + b} }$ | From [[Primitive of Power of x by Power of a x + b/Increment of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m +... | Primitive of Reciprocal of Power of x by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of x by Power of a x + b/Increment of Power of x",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9213 | Primitive of Power of x by Root of a x + b | :$\ds \int x^m \sqrt {a x + b} \rd x = \frac {2 x^m} {\paren {2 m + 3} a} \paren {\sqrt {a x + b} }^3 - \frac {2 m b} {\paren {2 m + 3} a} \int x^{m - 1} \sqrt{a x + b} \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x + b}^n \rd x$
Putting $n := \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \int ... | :$\ds \int x^m \sqrt {a x + b} \rd x = \frac {2 x^m} {\paren {2 m + 3} a} \paren {\sqrt {a x + b} }^3 - \frac {2 m b} {\paren {2 m + 3} a} \int x^{m - 1} \sqrt{a x + b} \rd x$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^m \paren {a x + b}^{n + 1} } {\paren {m + n + 1} a} - \frac {m b} {\paren {m + n + 1} a} \int x^{m - 1} \paren {a x... | Primitive of Power of x by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of x",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9214 | Primitive of Root of a x + b over Power of x/Formulation 1 | :$\ds \int \frac {\sqrt {a x + b} } {x^m} \rd x = -\frac {\sqrt {a x + b} } {\paren {m - 1} x^{m - 1} } + \frac a {2 \paren {m - 1} } \int \frac {\d x} {x^{m - 1} \sqrt {a x + b} }$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {2 \sqrt {a x + b} }
| c = Power Rule for Derivatives etc.
}}
{{eqn | l = v
| r = \frac {-1} {\paren {m - 1} x^{m - 1} }
| c =
}}
{{eqn | ll= \leadsto
... | :$\ds \int \frac {\sqrt {a x + b} } {x^m} \rd x = -\frac {\sqrt {a x + b} } {\paren {m - 1} x^{m - 1} } + \frac a {2 \paren {m - 1} } \int \frac {\d x} {x^{m - 1} \sqrt {a x + b} }$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac a {2 \sqrt {a x + b} }
| c = [[Power Rule for Derivatives]] etc.
}}
{{eqn | l = v
| r = \frac {-1} {\paren {m - 1} x^{m - 1} }
| c =
}}
{{eqn | ll= \lead... | Primitive of Root of a x + b over Power of x/Formulation 1 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_x/Formulation_1 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_x/Formulation_1 | [
"Primitive of Root of a x + b over Power of x"
] | [] | [
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9215 | Primitive of Root of a x + b over Power of x/Formulation 2 | :$\ds \int \frac {\sqrt{a x + b} } {x^m} \rd x = -\frac {\paren {\sqrt{a x + b} }^3} {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 5} a} {\paren {2 m - 2} b} \int \frac {\sqrt {a x + b} } {x^{m - 1} } \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$
Putting $n := \dfrac 1 2$ and $m := -m$:
{{... | :$\ds \int \frac {\sqrt{a x + b} } {x^m} \rd x = -\frac {\paren {\sqrt{a x + b} }^3} {\paren {m - 1} b x^{m - 1} } - \frac {\paren {2 m - 5} a} {\paren {2 m - 2} b} \int \frac {\sqrt {a x + b} } {x^{m - 1} } \rd x$ | From [[Primitive of Power of x by Power of a x + b/Increment of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m +... | Primitive of Root of a x + b over Power of x/Formulation 2 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_x/Formulation_2 | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_x/Formulation_2 | [
"Primitive of Root of a x + b over Power of x"
] | [] | [
"Primitive of Power of x by Power of a x + b/Increment of Power of x",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-9216 | Primitive of Power of Root of a x + b | :$\ds \int \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C$
for $m \ne -2$. | Let $u = \sqrt {a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int u \cdot u^m \rd x
| c = Primitive of Function of $\sqrt {a x + b}$
}}
{{eqn | r = \frac 2 a \int u^{m + 1} \rd x
| c = simplifying
}}
{{eqn | r = \frac 2 a \frac {u^{m + 2} } {m + 2} ... | :$\ds \int \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C$
for $m \ne -2$. | Let $u = \sqrt {a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int u \cdot u^m \rd x
| c = [[Primitive of Function of Root of a x + b|Primitive of Function of $\sqrt {a x + b}$]]
}}
{{eqn | r = \frac 2 a \int u^{m + 1} \rd x
| c = simplifying
}}
{{e... | Primitive of Power of Root of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b/Proof_1 | [
"Primitive of Power of Root of a x + b",
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of Root of a x + b",
"Primitive of Power"
] |
proofwiki-9217 | Primitive of x by Power of Root of a x + b | :$\ds \int x \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 4} } {a^2 \paren {m + 4} } - \frac {2 b \paren {\sqrt {a x + b} }^{m + 2} } {a^2 \paren {m + 2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int \frac {u^2 - b} a u^{m + 1} \rd x
| c = Prim... | :$\ds \int x \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 4} } {a^2 \paren {m + 4} } - \frac {2 b \paren {\sqrt {a x + b} }^{m + 2} } {a^2 \paren {m + 2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int \frac {u^2 - b} a u^{m + 1} \rd x
| c = [[... | Primitive of x by Power of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_by_Power_of_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9218 | Primitive of x squared by Power of Root of a x + b | :$\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^2 \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int \paren {\frac {u^2 - b} a}^2 u^{m + 1} \rd x
... | :$\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^2 \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int \paren {\frac {u^2 - b} a}^2 u^{m + 1} \rd x
... | Primitive of x squared by Power of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Power_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Power_of_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Function of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9219 | Primitive of Power of Root of a x + b over x | :$\ds \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x = \frac {2 \paren {\sqrt {a x + b} }^m } m + b \int \frac {\paren {\sqrt {a x + b} }^{m - 2} } x \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd x$
Putting $n := \dfrac m 2$ and $m := -1$:
{{begin-eqn}}
{{eqn | l = \int \fr... | :$\ds \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x = \frac {2 \paren {\sqrt {a x + b} }^m } m + b \int \frac {\paren {\sqrt {a x + b} }^{m - 2} } x \rd x$ | From [[Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b|Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $a x + b$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^n} {m + n + 1} + \frac {n b} {m + n + 1} \int x^m \paren {a x + b}^{n - 1} \rd... | Primitive of Power of Root of a x + b over x | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b_over_x | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b_over_x | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Decrement of Power of a x + b"
] |
proofwiki-9220 | Primitive of Power of Root of a x + b over x squared | :$\ds \int \frac {\paren {\sqrt {a x + b} }^m} {x^2} \rd x = -\frac {\paren {\sqrt {a x + b} }^{m + 2} } {b x} + \frac {m a} {2 b} \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x$ | From Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m + 1} \paren {a x + b}^n \rd x$
Putting $n := \dfrac m 2$ and $m := -2$:
{{... | :$\ds \int \frac {\paren {\sqrt {a x + b} }^m} {x^2} \rd x = -\frac {\paren {\sqrt {a x + b} }^{m + 2} } {b x} + \frac {m a} {2 b} \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x$ | From [[Primitive of Power of x by Power of a x + b/Increment of Power of x|Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $x$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {m + 1} b} - \frac {\paren {m + n + 2} a} {\paren {m + 1} b} \int x^{m +... | Primitive of Power of Root of a x + b over x squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b_over_x_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b_over_x_squared | [
"Primitives involving Root of a x + b"
] | [] | [
"Primitive of Power of x by Power of a x + b/Increment of Power of x"
] |
proofwiki-9221 | Primitive of Reciprocal of x by Power of Root of a x + b | :$\ds \int \frac {\d x} {x \paren {\sqrt {a x + b} }^m} = \frac 2 {\paren {m - 2} b \paren {\sqrt {a x + b} }^{m - 2} } + \frac 1 b \int \frac {\d x} {x \paren {\sqrt {a x + b} }^{m - 2} }$ | From Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $a x + b$:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m \paren {a x + b}^{n + 1} \rd x$
Putting $n := -\dfrac m 2$ and $m := -1$:
{{beg... | :$\ds \int \frac {\d x} {x \paren {\sqrt {a x + b} }^m} = \frac 2 {\paren {m - 2} b \paren {\sqrt {a x + b} }^{m - 2} } + \frac 1 b \int \frac {\d x} {x \paren {\sqrt {a x + b} }^{m - 2} }$ | From [[Primitive of Power of x by Power of a x + b/Increment of Power of a x + b|Primitive of Power of $x$ by Power of $a x + b$: Increment of Power of $a x + b$]]:
:$\ds \int x^m \paren {a x + b}^n \rd x = \frac {-x^{m + 1} \paren {a x + b}^{n + 1} } {\paren {n + 1} b} + \frac {m + n + 2} {\paren {n + 1} b} \int x^m ... | Primitive of Reciprocal of x by Power of Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b",
"Primitives involving Reciprocals"
] | [] | [
"Primitive of Power of x by Power of a x + b/Increment of Power of a x + b"
] |
proofwiki-9222 | Primitive of Reciprocal of a x + b by p x + q | :$\ds \int \frac {\d x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \ln \size {\frac {p x + q} {a x + b} } + C$
where $b p \ne a q$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b} \paren {p x + q} }
| r = \int \paren {\frac {-a} {\paren {b p - a q} \paren {a x + b} } + \frac p {\paren {b p - a q} \paren {p x + q} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b p - a q} \paren { {-a} \int \frac 1 {a x +... | :$\ds \int \frac {\d x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \ln \size {\frac {p x + q} {a x + b} } + C$
where $b p \ne a q$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b} \paren {p x + q} }
| r = \int \paren {\frac {-a} {\paren {b p - a q} \paren {a x + b} } + \frac p {\paren {b p - a q} \paren {p x + q} } } \rd x
| c = [[Primitive of Reciprocal of a x + b by p x + q/Partial Fraction Expansion|Partial Fraction Exp... | Primitive of Reciprocal of a x + b by p x + q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_by_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_by_p_x_+_q | [
"Primitive of Reciprocal of a x + b by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of Reciprocal of a x + b by p x + q/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-9223 | Primitive of x over a x + b by p x + q | :$\ds \int \frac {x \rd x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac b a \ln \size {a x + b} - \frac q p \ln \size {p x + q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b} \paren {p x + q} }
| r = \int \paren {\frac b {\paren {b p - a q} \paren {a x + b} } - \frac q {\paren {b p - a q} \paren {p x + q} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b p - a q} \paren {b \int \frac 1 {a x + b} ... | :$\ds \int \frac {x \rd x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac b a \ln \size {a x + b} - \frac q p \ln \size {p x + q} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b} \paren {p x + q} }
| r = \int \paren {\frac b {\paren {b p - a q} \paren {a x + b} } - \frac q {\paren {b p - a q} \paren {p x + q} } } \rd x
| c = [[Primitive of x over a x + b by p x + q/Partial Fraction Expansion|Partial Fraction Expansion]... | Primitive of x over a x + b by p x + q | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_by_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_by_p_x_+_q | [
"Primitive of x over a x + b by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of x over a x + b by p x + q/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9224 | Primitive of Reciprocal of a x + b squared by p x + q | :$\ds \int \frac {\d x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac 1 {a x + b} + \frac p {b p - a q} \ln \size {\frac {p x + q} {a x + b} } } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac 1 {b p - a q} \paren {\frac {-a p} {\paren {b p - a q} \paren {a x + b} } + \frac {-a} {\paren {a x + b}^2} + \frac {p^2} {\paren {b p - a q} \paren {p x + q} } } } \rd x
... | :$\ds \int \frac {\d x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac 1 {a x + b} + \frac p {b p - a q} \ln \size {\frac {p x + q} {a x + b} } } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac 1 {b p - a q} \paren {\frac {-a p} {\paren {b p - a q} \paren {a x + b} } + \frac {-a} {\paren {a x + b}^2} + \frac {p^2} {\paren {b p - a q} \paren {p x + q} } } } \rd x
... | Primitive of Reciprocal of a x + b squared by p x + q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared_by_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared_by_p_x_+_q | [
"Primitive of Reciprocal of a x + b squared by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of Reciprocal of a x + b squared by p x + q/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9225 | Primitive of x over a x + b squared by p x + q | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac 1 {b p - a q} \paren {\frac {a q} {\paren {b p - a q} \paren {a x + b} } + \frac b {\paren {a x + b}^2} - \frac {p q} {\paren {b p - a q} \paren {p x + q} } } } \rd x
|... | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac 1 {b p - a q} \paren {\frac {a q} {\paren {b p - a q} \paren {a x + b} } + \frac b {\paren {a x + b}^2} - \frac {p q} {\paren {b p - a q} \paren {p x + q} } } } \rd x
|... | Primitive of x over a x + b squared by p x + q | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared_by_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared_by_p_x_+_q | [
"Primitive of x over a x + b squared by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of x over a x + b squared by p x + q/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-9226 | Primitive of x squared over a x + b squared by p x + q | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac {-b^2} {a \paren {b p - a q} \paren {a x + b}^2} + \frac {q^2} {\paren {b p - a q}^2 \paren {p x + q} } + \frac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2 \paren {a ... | :$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C$ | {{begin-eqn}}
{{eqn | o =
| r = \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| c =
}}
{{eqn | r = \int \paren {\frac {-b^2} {a \paren {b p - a q} \paren {a x + b}^2} + \frac {q^2} {\paren {b p - a q}^2 \paren {p x + q} } + \frac {b \paren {b p - 2 a q} } {a \paren {b p - a q}^2 \paren {a ... | Primitive of x squared over a x + b squared by p x + q | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared_by_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_x_squared_over_a_x_+_b_squared_by_p_x_+_q | [
"Primitive of x squared over a x + b squared by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of x squared over a x + b squared by p x + q/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of a x + b squared",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b"
] |
proofwiki-9227 | Friedrichs' Inequality | Let $G \subset \R^n$ be bounded domain.
Then for any $u \in \map {W^{1, 2}_0} G$:
:$\norm u_{\map {L^2} G} \le \map {\operatorname {diam} } G \norm {\nabla u}_{\map {L^2} G}$
where:
:$\map {\operatorname {diam} } G := \sup \limits_{x, y \mathop \in G} \size {x - y}$ | === Smooth functions with compact support ===
{{explain|In several places in the below, this construct was seen: $\vert \nabla u (x_1, \dots, x_{m - 1}, t\vert^2$ where it appears that the close of the round bracket parenthesis "$)$" was been omitted. It has been assmued in the corrected version that it should be this:... | Let $G \subset \R^n$ be bounded domain.
Then for any $u \in \map {W^{1, 2}_0} G$:
:$\norm u_{\map {L^2} G} \le \map {\operatorname {diam} } G \norm {\nabla u}_{\map {L^2} G}$
where:
:$\map {\operatorname {diam} } G := \sup \limits_{x, y \mathop \in G} \size {x - y}$ | === Smooth functions with compact support ===
{{explain|In several places in the below, this construct was seen: $\vert \nabla u (x_1, \dots, x_{m - 1}, t\vert^2$ where it appears that the close of the round bracket parenthesis "$)$" was been omitted. It has been assmued in the corrected version that it should be this... | Friedrichs' Inequality | https://proofwiki.org/wiki/Friedrichs'_Inequality | https://proofwiki.org/wiki/Friedrichs'_Inequality | [
"Sobolev Spaces"
] | [] | [
"Cauchy-Bunyakovsky-Schwarz Inequality",
"Fubini's Theorem"
] |
proofwiki-9228 | Primitive of Reciprocal of Power of a x + b by Power of p x + q | :$\ds \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^n} = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac 1 {\paren {a x + b}^{m - 1} \paren {p x + q}^{n - 1} } + a \paren {m + n - 2} \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^{n - 1} } }$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | :$\ds \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^n} = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac 1 {\paren {a x + b}^{m - 1} \paren {p x + q}^{n - 1} } + a \paren {m + n - 2} \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^{n - 1} } }$ | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \p... | Primitive of Reciprocal of Power of a x + b by Power of p x + q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q | [
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9229 | Primitive of Power of a x + b by Power of p x + q/Decrement of Power | :$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd x$ | 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 = \paren {a x + b}^s
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
| r = a... | :$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd x$ | 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 = \paren {a x + b}^s
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d v} {\d x}
|... | Primitive of Power of a x + b by Power of p x + q/Decrement of Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q/Decrement_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q/Decrement_of_Power | [
"Primitive of Power of a x + b by Power of p x + q"
] | [] | [
"Integration by Parts",
"Power Rule for Derivatives",
"Derivative of Function of Constant Multiple/Corollary",
"Definition:Integration/Integrand",
"Product Rule for Derivatives",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9230 | Primitive of Power of a x + b by Power of p x + q/Increment of Power | :$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | :$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\... | Primitive of Power of a x + b by Power of p x + q/Increment of Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q/Increment_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_by_Power_of_p_x_+_q/Increment_of_Power | [
"Primitive of Power of a x + b by Power of p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9231 | Primitive of a x + b over p x + q | :$\ds \int \frac {a x + b} {p x + q} \rd x = \frac {a x} p + \frac {b p - a q} {p^2} \ln \size {p x + q} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {a x + b} {p x + q} \rd x
| r = a \int \frac {x \rd x} {p x + q} + b \int \frac {\d x} {p x + q}
| c = Linear Combination of Primitives
}}
{{eqn | r = a \paren {\frac x p - \frac q {p^2} \ln \size {p x + q} } + b \int \frac {\d x} {p x + q} + C
| c = Primitive of $... | :$\ds \int \frac {a x + b} {p x + q} \rd x = \frac {a x} p + \frac {b p - a q} {p^2} \ln \size {p x + q} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {a x + b} {p x + q} \rd x
| r = a \int \frac {x \rd x} {p x + q} + b \int \frac {\d x} {p x + q}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = a \paren {\frac x p - \frac q {p^2} \ln \size {p x + q} } + b \int \frac {\d x} {p x + q} + C
| c = [[Primitiv... | Primitive of a x + b over p x + q | https://proofwiki.org/wiki/Primitive_of_a_x_+_b_over_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_a_x_+_b_over_p_x_+_q | [
"Primitives involving a x + b and p x + q"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over a x + b",
"Primitive of Reciprocal of a x + b",
"Definition:Common Denominator"
] |
proofwiki-9232 | Primitive of Power of a x + b over Power of p x + q/Formulation 1 | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac {\paren {a x + b}^{m + 1} } {\paren {p x + q}^{n - 1} } + \paren {n - m - 2} a \int \frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } \rd x}$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - \paren {m + n + 2} a \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
... | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac {\paren {a x + b}^{m + 1} } {\paren {p x + q}^{n - 1} } + \paren {n - m - 2} a \int \frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } \rd x}$ | From [[Primitive of Power of a x + b by Power of p x + q/Increment of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Increment of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - \par... | Primitive of Power of a x + b over Power of p x + q/Formulation 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_1 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_1 | [
"Primitive of Power of a x + b over Power of p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Increment of Power"
] |
proofwiki-9233 | Primitive of Power of a x + b over Power of p x + q/Formulation 2 | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \paren {b p - a q} \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^n} \rd x}$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^m \paren {p x + q}^{n + 1} } {\paren {m + n + 1} a} + \frac {m \paren {b p - a q} } {\paren {m + n + 1} p} \int \paren {a x + b}^{m - 1} \paren {p x + q}^n \rd... | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \paren {b p - a q} \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^n} \rd x}$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^m \paren {p x + q}^{n + 1} } {\paren {m + n + 1} a} + \frac {m \paren {b p - a q} } {... | Primitive of Power of a x + b over Power of p x + q/Formulation 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_2 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_2 | [
"Primitive of Power of a x + b over Power of p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9234 | Primitive of Power of a x + b over Power of p x + q/Formulation 3 | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } - m a \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^{n - 1}} \rd x}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \paren {a x + b}^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m a \paren {a x + b}^m
| c = Power Rule for Derivatives and Derivatives of Function of $a x + b$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = v
| r = \frac {\par... | :$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } - m a \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^{n - 1}} \rd x}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \paren {a x + b}^m
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = m a \paren {a x + b}^m
| c = [[Power Rule for Derivatives]] and [[Derivatives of Function of a x + b|Derivatives of Function of $a x + b$]]
}}
{{end-eqn}}
Then:
{{beg... | Primitive of Power of a x + b over Power of p x + q/Formulation 3 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_3 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b_over_Power_of_p_x_+_q/Formulation_3 | [
"Primitive of Power of a x + b over Power of p x + q"
] | [] | [
"Power Rule for Derivatives",
"Derivatives of Function of a x + b",
"Power Rule for Derivatives",
"Derivatives of Function of a x + b",
"Integration by Parts"
] |
proofwiki-9235 | Primitive of p x + q over Root of a x + b | :$\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt{a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {p x + q} {\sqrt {a x + b} } \rd x
| r = \int \paren {p \paren {\frac {u^2 - b} a} + q} \frac {2 u} {a... | :$\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt{a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {p x + q} {\sqrt {a x + b} } \rd x
| r = \int \paren {p \paren {\frac {u^2 - b} a} + q} \frac {2 u} ... | Primitive of p x + q over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_p_x_+_q_over_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Function of Root of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power"
] |
proofwiki-9236 | Primitive of Reciprocal of p x + q by Root of a x + b | Let $a, b, p, q \in \R$ such that $a p \ne b q$ and such that $p \ne 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \begin {cases}
\dfrac 1 {\sqrt {p \paren {b p - a q} } } \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a... | === Case $1$: $p \paren {b p - a q} > 0$ ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0}}{{qed|lemma}} | Let $a, b, p, q \in \R$ such that $a p \ne b q$ and such that $p \ne 0$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \begin {cases}
\dfrac 1 {\sqrt {p \paren {b p - a q} } } \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p -... | === [[Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0|Case $1$: $p \paren {b p - a q} > 0$]] ===
{{:Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0}}{{qed|lemma}} | Primitive of Reciprocal of p x + q by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b | [
"Primitive of Reciprocal of p x + q by Root of a x + b",
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) greater than 0"
] |
proofwiki-9237 | Primitive of Root of a x + b over p x + q | :<nowiki>$\ds \int \frac {\sqrt{a x + b} } {p x + q} \rd x = \begin{cases}
\dfrac {2 \sqrt{a x + b} } p + \dfrac {\sqrt {b p - a q} } {p \sqrt p} \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } & : b p - a q > 0 \\ \\
\dfrac {2 \sqrt{a x + b} } p... | From Primitive of Power of $a x + b$ over Power of $p x + q$: Formulation 2:
:$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \paren {b p - a q} \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}... | :<nowiki>$\ds \int \frac {\sqrt{a x + b} } {p x + q} \rd x = \begin{cases}
\dfrac {2 \sqrt{a x + b} } p + \dfrac {\sqrt {b p - a q} } {p \sqrt p} \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } & : b p - a q > 0 \\ \\
\dfrac {2 \sqrt{a x + b} } p... | From [[Primitive of Power of a x + b over Power of p x + q/Formulation 2|Primitive of Power of $a x + b$ over Power of $p x + q$: Formulation 2]]:
:$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \par... | Primitive of Root of a x + b over p x + q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_p_x_+_q | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Power of a x + b over Power of p x + q/Formulation 2",
"Primitive of Reciprocal of p x + q by Root of a x + b"
] |
proofwiki-9238 | Primitive of Power of p x + q by Root of a x + b | :$\ds \int \paren {p x + q}^n \sqrt {a x + b} \rd x = \frac {2 \paren {p x + q}^{n + 1} \sqrt {a x + b} } {\paren {2 n + 3} p} + \frac {b p - a q} {\paren {2 n + 3} p} \int \frac {\paren {p x + q}^n} {\sqrt{a x + b} } \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^m \paren {p x + q}^{n + 1} } {\paren {m + n + 1} p} + \frac {m \paren {b p - a q} } {\paren {m + n + 1} p} \int \paren {a x + b}^{m - 1} \paren {p x + q}^n \rd... | :$\ds \int \paren {p x + q}^n \sqrt {a x + b} \rd x = \frac {2 \paren {p x + q}^{n + 1} \sqrt {a x + b} } {\paren {2 n + 3} p} + \frac {b p - a q} {\paren {2 n + 3} p} \int \frac {\paren {p x + q}^n} {\sqrt{a x + b} } \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^m \paren {p x + q}^{n + 1} } {\paren {m + n + 1} p} + \frac {m \paren {b p - a q} } {... | Primitive of Power of p x + q by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Power_of_p_x_+_q_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_p_x_+_q_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9239 | Primitive of Reciprocal of Power of p x + q by Root of a x + b | :$\ds \int \frac {\d x} {\paren {p x + q}^n \sqrt {a x + b} } = \frac {\sqrt {a x + b} } {\paren {n - 1} \paren {a q - b p} \paren {p x + q}^{n - 1} } + \frac {\paren {2 n - 3} a} {2 \paren {n - 1} \paren {a q - b p} } \int \frac {\d x} {\paren {p x + q}^{n - 1} \sqrt {a x + b} }$ | From Primitive of Reciprocal of $\dfrac 1 {\paren {a x + b}^m \paren {p x + q}^n}$:
:$\ds \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^n} = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac 1 {\paren {a x + b}^{m - 1} \paren {p x + q}^{n - 1} } + a \paren {m + n - 2} \int \frac {\d x} {\paren {a x... | :$\ds \int \frac {\d x} {\paren {p x + q}^n \sqrt {a x + b} } = \frac {\sqrt {a x + b} } {\paren {n - 1} \paren {a q - b p} \paren {p x + q}^{n - 1} } + \frac {\paren {2 n - 3} a} {2 \paren {n - 1} \paren {a q - b p} } \int \frac {\d x} {\paren {p x + q}^{n - 1} \sqrt {a x + b} }$ | From [[Primitive of Reciprocal of Power of a x + b by Power of p x + q|Primitive of Reciprocal of $\dfrac 1 {\paren {a x + b}^m \paren {p x + q}^n}$]]:
:$\ds \int \frac {\d x} {\paren {a x + b}^m \paren {p x + q}^n} = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac 1 {\paren {a x + b}^{m - 1} \paren {p x... | Primitive of Reciprocal of Power of p x + q by Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_p_x_+_q_by_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_p_x_+_q_by_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Reciprocal of Power of a x + b by Power of p x + q"
] |
proofwiki-9240 | Primitive of Power of p x + q over Root of a x + b | :$\ds \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x = \frac {2 \paren {p x + q}^n \sqrt {a x + b} } {\paren {2 n + 1} a} + \frac {2 n \paren {a q - b p} } {\paren {2 n + 1} a} \int \frac {\paren {p x + q}^{n - 1} } {\sqrt {a x + b} } \rd x$ | From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd ... | :$\ds \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x = \frac {2 \paren {p x + q}^n \sqrt {a x + b} } {\paren {2 n + 1} a} + \frac {2 n \paren {a q - b p} } {\paren {2 n + 1} a} \int \frac {\paren {p x + q}^{n - 1} } {\sqrt {a x + b} } \rd x$ | From [[Primitive of Power of a x + b by Power of p x + q/Decrement of Power|Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power]]:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\... | Primitive of Power of p x + q over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Power_of_p_x_+_q_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Power_of_p_x_+_q_over_Root_of_a_x_+_b | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Power of a x + b by Power of p x + q/Decrement of Power"
] |
proofwiki-9241 | Primitive of Root of a x + b over Power of p x + q | :$\ds \int \frac {\sqrt {a x + b} } {\paren {p x + q}^n} \rd x = \frac {-\sqrt {a x + b} } {\paren {n - 1} p \paren {p x + q}^{n - 1} } + \frac a {2 \paren {n - 1} p} \int \frac {\d x} {\paren {p x + q}^{n - 1} \sqrt {a x + b} }$ | From Primitive of Power of $a x + b$ over Power of $p x + q$: Formulation 3:
:$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m a \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^{n - 1} } \rd x}$
Se... | :$\ds \int \frac {\sqrt {a x + b} } {\paren {p x + q}^n} \rd x = \frac {-\sqrt {a x + b} } {\paren {n - 1} p \paren {p x + q}^{n - 1} } + \frac a {2 \paren {n - 1} p} \int \frac {\d x} {\paren {p x + q}^{n - 1} \sqrt {a x + b} }$ | From [[Primitive of Power of a x + b over Power of p x + q/Formulation 3|Primitive of Power of $a x + b$ over Power of $p x + q$: Formulation 3]]:
:$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m a \int \... | Primitive of Root of a x + b over Power of p x + q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_over_Power_of_p_x_+_q | [
"Primitives involving Root of a x + b and p x + q"
] | [] | [
"Primitive of Power of a x + b over Power of p x + q/Formulation 3"
] |
proofwiki-9242 | Primitive of Reciprocal of Root of a x + b by Root of p x + q | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases}
\dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C & : a p < 0 \\ \\
\dfrac {-1} {\sqrt {-a p} } \map \arcsin {\dfr... | === Case $1$: $a p < 0$ ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0}}{{qed|lemma}} | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases}
\dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C & : a p < 0 \\ \\
\dfrac {-1} {\sqrt {-a p} } \map \arcsin {\df... | === [[Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0|Case $1$: $a p < 0$]] ===
{{:Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0}}{{qed|lemma}} | Primitive of Reciprocal of Root of a x + b by Root of p x + q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_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",
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0"
] |
proofwiki-9243 | Primitive of Reciprocal of Root of a x + b by Root of p x + q | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases}
\dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C & : a p < 0 \\ \\
\dfrac {-1} {\sqrt {-a p} } \map \arcsin {\dfr... | First let us express the integrand in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{end-eqn}}
Recall the defin... | Let $a, b, p, q \in \R$ such that $a p \ne b q$.
Then:
:<nowiki>$\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \begin {cases}
\dfrac 2 {\sqrt {-a p} } \map \arctan {\sqrt {\dfrac {-p \paren {a x + b} } {a \paren {p x + q} } } } + C & : a p < 0 \\ \\
\dfrac {-1} {\sqrt {-a p} } \map \arcsin {\df... | First let us express the [[Definition:Integrand|integrand]] in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | [
"Primitive of Reciprocal of Root of a x + b by Root of p x + q",
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Euler Substitution/Third",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arccotangent of Reciprocal equals Arctangent",
"Sum of A... |
proofwiki-9244 | Derivative of Laplace Transform | Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\closedint 0 a$.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then, everywhere that $\dfrac \d {\d s} \laptrans f$ exists:
:$\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$ | {{mistake|see talk page}}
{{begin-eqn}}
{{eqn | l = \frac \d {\d s} \laptrans {\map f t}
| r = \frac \d {\d s} \int_0^{\to +\infty} \map f t \, e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \map {\frac {\partial} {\partial s} } {\map f t \, e^{-s t} } \rd t
| c =... | Let $f: \R \to \R$ or $\R \to \C$ be a [[Definition:Continuous|continuous]] [[Definition:Function|function]], [[Definition:Differentiable on Interval|differentiable]] on any [[Definition:Closed Real Interval|closed interval]] $\closedint 0 a$.
Let $\laptrans f = F$ denote the [[Definition:Laplace Transform|Laplace tra... | {{mistake|see talk page}}
{{begin-eqn}}
{{eqn | l = \frac \d {\d s} \laptrans {\map f t}
| r = \frac \d {\d s} \int_0^{\to +\infty} \map f t \, e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \map {\frac {\partial} {\partial s} } {\map f t \, e^{-s t} } \rd t
| c ... | Derivative of Laplace Transform | https://proofwiki.org/wiki/Derivative_of_Laplace_Transform | https://proofwiki.org/wiki/Derivative_of_Laplace_Transform | [
"Derivatives of Laplace Transforms",
"Laplace Transforms",
"Derivatives"
] | [
"Definition:Continuous",
"Definition:Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Laplace Transform"
] | [
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function"
] |
proofwiki-9245 | Primitive of x over Root of a x + b by Root of p x + q | :$\ds \int \frac {x \rd x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \frac {\sqrt {\paren {a x + b} \paren {p x + q} } } {a p} - \frac {b p + a q} {2 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {p x + q}
| r = \sqrt {p \paren {\frac {u^2 - b} a} + q}
| c =
}}
{{eqn | r = \sqrt {\frac {p \paren {u^2 - b}... | :$\ds \int \frac {x \rd x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \frac {\sqrt {\paren {a x + b} \paren {p x + q} } } {a p} - \frac {b p + a q} {2 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \sqrt {a x + b}
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \frac {u^2 - b} a
| c =
}}
{{eqn | ll= \leadsto
| l = \sqrt {p x + q}
| r = \sqrt {p \paren {\frac {u^2 - b} a} + q}
| c =
}}
{{eqn | r = \sqrt {\frac {p \paren {u^2 - b}... | Primitive of x over Root of a x + b by Root of p x + q | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_+_b_by_Root_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_x_over_Root_of_a_x_+_b_by_Root_of_p_x_+_q | [
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Primitive of Function of Root of a x + b",
"Linear Combination of Integrals/Indefinite",
"Primitive of x squared over Root of x squared minus a squared",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Definition:Common Denominator",
"Primitive of Reciprocal of Root of a x... |
proofwiki-9246 | Higher Order Derivatives of Laplace Transform | :$\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$ | The proof proceeds by induction on $n$, the order of the derivative of $\laptrans f$.
The case $n = 0$ is verified as follows:
{{begin-eqn}}
{{eqn | l = \frac {\d^0} {\d s^0} \laptrans {\map f t}
| r = \laptrans {\map f t}
| c = {{Defof|Zeroth Derivative}}
}}
{{eqn | r = \paren {-1}^0 \laptrans {t^0 \, \map... | :$\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$ | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$, the [[Definition:Order of Derivative|order of the derivative]] of $\laptrans f$.
The case $n = 0$ is verified as follows:
{{begin-eqn}}
{{eqn | l = \frac {\d^0} {\d s^0} \laptrans {\map f t}
| r = \laptrans {\map f t}
| c = {... | Higher Order Derivatives of Laplace Transform | https://proofwiki.org/wiki/Higher_Order_Derivatives_of_Laplace_Transform | https://proofwiki.org/wiki/Higher_Order_Derivatives_of_Laplace_Transform | [
"Derivatives of Laplace Transforms",
"Laplace Transforms",
"Derivatives"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Derivative/Higher Derivatives/Order of Derivative",
"Principle of Mathematical Induction"
] |
proofwiki-9247 | Primitive of Root of a x + b by Root of p x + q | :$\ds \int \sqrt {\paren {a x + b} \paren {p x + q} } \rd x = \frac {2 a p x + b p + a q} {4 a p} \sqrt {\paren {a x + b} \paren {p x + q} } - \frac {\paren {b p - a q}^2} {8 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | From Primitive of $\paren {p x + q}^n \sqrt {a x + b}$:
:$\ds \int \paren {p x + q}^n \sqrt {a x + b} \rd x = \frac {2 \paren {p x + q}^{n + 1} \sqrt {a x + b} } {\paren {2 n + 3} p} + \frac {b p - a q} {\paren {2 n + 3} p} \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x$
Putting $n = \dfrac 1 2$:
{{begin-eqn}... | :$\ds \int \sqrt {\paren {a x + b} \paren {p x + q} } \rd x = \frac {2 a p x + b p + a q} {4 a p} \sqrt {\paren {a x + b} \paren {p x + q} } - \frac {\paren {b p - a q}^2} {8 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | From [[Primitive of Power of p x + q by Root of a x + b|Primitive of $\paren {p x + q}^n \sqrt {a x + b}$]]:
:$\ds \int \paren {p x + q}^n \sqrt {a x + b} \rd x = \frac {2 \paren {p x + q}^{n + 1} \sqrt {a x + b} } {\paren {2 n + 3} p} + \frac {b p - a q} {\paren {2 n + 3} p} \int \frac {\paren {p x + q}^n} {\sqrt {a x... | Primitive of Root of a x + b by Root of p x + q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q | [
"Primitive of Root of a x + b by Root of p x + q",
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Primitive of Power of p x + q by Root of a x + b",
"Primitive of Root of p x + q over Root of a x + b"
] |
proofwiki-9248 | Primitive of Root of p x + q over Root of a x + b | :$\ds \int \sqrt {\frac {p x + q} {a x + b} } \rd x = \frac {\sqrt {\paren {a x + b} \paren {p x + q} } } a + \frac {a q - b p} {2 a} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | From Primitive of $\dfrac {\paren {p x + q}^n} {\sqrt {a x + b} }$:
:$\ds \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x = \frac {2 \paren {p x + q}^n \sqrt {a x + b} } {\paren {2 n + 1} a} + \frac {2 n \paren {a q - b p} } {\paren {2 n + 1} a} \int \frac {\paren {p x + q}^{n - 1} } {\sqrt {a x + b} } \rd x$
... | :$\ds \int \sqrt {\frac {p x + q} {a x + b} } \rd x = \frac {\sqrt {\paren {a x + b} \paren {p x + q} } } a + \frac {a q - b p} {2 a} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$ | From [[Primitive of Power of p x + q over Root of a x + b|Primitive of $\dfrac {\paren {p x + q}^n} {\sqrt {a x + b} }$]]:
:$\ds \int \frac {\paren {p x + q}^n} {\sqrt {a x + b} } \rd x = \frac {2 \paren {p x + q}^n \sqrt {a x + b} } {\paren {2 n + 1} a} + \frac {2 n \paren {a q - b p} } {\paren {2 n + 1} a} \int \frac... | Primitive of Root of p x + q over Root of a x + b | https://proofwiki.org/wiki/Primitive_of_Root_of_p_x_+_q_over_Root_of_a_x_+_b | https://proofwiki.org/wiki/Primitive_of_Root_of_p_x_+_q_over_Root_of_a_x_+_b | [
"Primitive of Root of p x + q over Root of a x + b",
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Primitive of Power of p x + q over Root of a x + b"
] |
proofwiki-9249 | Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q | :$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } } = \frac {2 \sqrt {a x + b} } {\paren {a q - b p} \sqrt {p x + q} } + C$ | From Primitive of $\paren {p x + q}^n \sqrt {a x + b}$:
:$\ds \int \frac {\d x} {\paren {p x + q}^n \sqrt {a x + b} } = \frac {\sqrt {a x + b} } {\paren {n - 1} \paren {a q - b p} \paren {p x + q}^{n - 1} } + \frac {\paren {2 n - 3} a} {2 \paren {n - 1} \paren {a q - b p} } \int \frac {\d x} {\paren {p x + q}^{n - 1} }... | :$\ds \int \frac {\d x} {\paren {p x + q} \sqrt {\paren {a x + b} \paren {p x + q} } } = \frac {2 \sqrt {a x + b} } {\paren {a q - b p} \sqrt {p x + q} } + C$ | From [[Primitive of Reciprocal of Power of p x + q by Root of a x + b|Primitive of $\paren {p x + q}^n \sqrt {a x + b}$]]:
:$\ds \int \frac {\d x} {\paren {p x + q}^n \sqrt {a x + b} } = \frac {\sqrt {a x + b} } {\paren {n - 1} \paren {a q - b p} \paren {p x + q}^{n - 1} } + \frac {\paren {2 n - 3} a} {2 \paren {n - 1}... | Primitive of Reciprocal of p x + q by Root of a x + b by Root of p x + q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b_by_Root_of_p_x_+_q | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_x_+_q_by_Root_of_a_x_+_b_by_Root_of_p_x_+_q | [
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Primitive of Reciprocal of Power of p x + q by Root of a x + b"
] |
proofwiki-9250 | Primitive of x over x squared plus a squared | :$\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^2 + a^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Power Rule for Derivatives and Derivative of Constant
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^2 + a^2}
| r = \frac 1 2 \ln \size {x^2 + a^2} + C
... | :$\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = u
| r = x^2 + a^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]] and [[Derivative of Constant]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {x \rd x} {x^2 + a^2}
| r = \frac 1 2 \ln \size {x^2 + a^... | Primitive of x over x squared plus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared/Proof_1 | [
"Primitive of x over x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Primitive of Function under its Derivative",
"Absolute Value of Even Power"
] |
proofwiki-9251 | Primitive of x over x squared plus a squared | :$\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$ | From Primitive of Power of x less one over Power of x plus Power of a:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {x^2 + a^2}
| r = \frac 1 2 \ln \size {x^2 + a^2} + C
| c = Primitive of $\dfrac {x^{n - 1} } {\pare... | :$\ds \int \frac {x \rd x} {x^2 + a^2} = \frac 1 2 \map \ln {x^2 + a^2} + C$ | From [[Primitive of Power of x less one over Power of x plus Power of a]]:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {x^2 + a^2}
| r = \frac 1 2 \ln \size {x^2 + a^2} + C
| c = [[Primitive of Power of x less one... | Primitive of x over x squared plus a squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared/Proof_2 | [
"Primitive of x over x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Power of x less one over Power of x plus Power of a",
"Primitive of Power of x less one over Power of x plus Power of a",
"Absolute Value of Even Power"
] |
proofwiki-9252 | Primitive of x squared over x squared plus a squared | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 + a^2}
| r = \int \paren {1 - \frac {a^2} {x^2 + a^2} } \rd x
| c = long division
}}
{{eqn | r = \int \d x - a^2 \int \frac {\d x} {x^2 + a^2}
| c = Linear Combination of Primitives
}}
{{eqn | r = x - a^2 \int \frac {\d x} {x^2 + a^2} + C
| ... | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {x^2 + a^2}
| r = \int \paren {1 - \frac {a^2} {x^2 + a^2} } \rd x
| c = long division
}}
{{eqn | r = \int \d x - a^2 \int \frac {\d x} {x^2 + a^2}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r = x - a^2 \int \frac {\d x} {x^2 + a^2} + C
... | Primitive of x squared over x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Constant",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9253 | Primitive of x squared over x squared plus a squared | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {a^2 \tan^2... | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {a^2 \t... | Primitive of x squared over x squared plus a squared squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared/Proof_1 | [
"Primitives involving x squared plus a squared"
] | [] | [
"Derivative of Tangent Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Square of Sine Function",
"Tangent Half-Angle Substitution for Sine"
] |
proofwiki-9254 | Primitive of x squared over x squared plus a squared | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}
| c = Linear Combination of Primitives
}}
{{e... | :$\ds \int \frac {x^2 \rd x} {x^2 + a^2} = x - a \arctan {\frac x a} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}
| c = [[Linear Combination of Primitives]]
}}... | Primitive of x squared over x squared plus a squared squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared/Proof_2 | [
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared plus a squared squared"
] |
proofwiki-9255 | Primitive of x cubed over x squared plus a squared | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^2 + a^2}
| r = \int \paren {x - \frac {a^2 x} {x^2 + a^2} } \rd x
| c = Polynomial Division
}}
{{eqn | r = \int x \rd x - a^2 \int \frac {x \rd x} {x^2 + a^2}
| c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x^2} 2 - a^2 \int \frac {x \r... | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^2 + a^2}
| r = \int \paren {x - \frac {a^2 x} {x^2 + a^2} } \rd x
| c = [[Definition:Polynomial Division|Polynomial Division]]
}}
{{eqn | r = \int x \rd x - a^2 \int \frac {x \rd x} {x^2 + a^2}
| c = [[Linear Combination of Primitives]]
}}
{{eqn | r... | Primitive of x cubed over x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Definition:Polynomial Division",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of x over x squared plus a squared"
] |
proofwiki-9256 | Primitive of x cubed over x squared plus a squared | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {\paren {z - a^2} } {z^2} \frac {\d z} 2
... | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {\paren {z - a^2} } {z^2} \frac {\d z} ... | Primitive of x cubed over x squared plus a squared squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared/Proof_1 | [
"Primitives involving x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal"
] |
proofwiki-9257 | Primitive of x cubed over x squared plus a squared | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x \paren {x^2 + a^2 - a^2} } {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2} } {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}
| c = Linear Combi... | :$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x \paren {x^2 + a^2 - a^2} } {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2} } {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}
| c = [[Linear Com... | Primitive of x cubed over x squared plus a squared squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared/Proof_2 | [
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over x squared plus a squared",
"Primitive of x over x squared minus a squared squared"
] |
proofwiki-9258 | Primitive of Reciprocal of x by x squared plus a squared | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x} - \frac x {a^2 \paren {x^2 + a^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} {x^2 + a^2}
| c = Linear Combination... | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x} - \frac x {a^2 \paren {x^2 + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x by x squared plus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_1 | [
"Primitive of Reciprocal of x by x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Difference of Logarithms"
] |
proofwiki-9259 | Primitive of Reciprocal of x by x squared plus a squared | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }
| c = multiplying top and bottom by $a^2$
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x \paren {x^2 + a^2} }
| c = adding and subtracting $x^2$
}}
{{eqn | r ... | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^2$
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x \paren {x^2 + a^2} }
... | Primitive of Reciprocal of x by x squared plus a squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_2 | [
"Primitive of Reciprocal of x by x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Difference of Logarithms"
] |
proofwiki-9260 | Primitive of Reciprocal of x by x squared plus a squared | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | From Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \frac 1 {2 a^2} \ln \size {\frac {x^2} {x^2 + a^2} } + C
| c = Primi... | :$\ds \int \frac {\rd x} {x \paren {x^2 + a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | From [[Primitive of Reciprocal of x by Power of x plus Power of a|Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$]]:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \frac 1 {... | Primitive of Reciprocal of x by x squared plus a squared/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_3 | [
"Primitive of Reciprocal of x by x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Absolute Value of Even Power"
] |
proofwiki-9261 | Primitive of Reciprocal of x squared by x squared plus a squared | :$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2} } = -\frac 1 {a^2 x} - \frac 1 {a^3} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x^2} - \frac 1 {a^2 \paren {x^2 + a^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} {x^2} - \frac 1 {a^2} \int \frac {1 \rd x} {x^2 + a^2}
| c = Linear Com... | :$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2} } = -\frac 1 {a^2 x} - \frac 1 {a^3} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x^2} - \frac 1 {a^2 \paren {x^2 + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x squared by x squared plus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2}... | Primitive of Reciprocal of x squared by x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_plus_a_squared | [
"Primitive of Reciprocal of x squared by x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x squared by x squared plus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-9262 | Primitive of Reciprocal of x cubed by x squared plus a squared | :$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2} } = -\frac 1 {2 a^2 x^2} - \frac 1 {2 a^4} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x^3} - \frac 1 {a^4 x} + \frac x {a^4 \paren {x^2 + a^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} \... | :$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2} } = -\frac 1 {2 a^2 x^2} - \frac 1 {2 a^4} \map \ln {\frac {x^2 + a^2} {x^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x^3} - \frac 1 {a^4 x} + \frac x {a^4 \paren {x^2 + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x cubed by x squared plus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r... | Primitive of Reciprocal of x cubed by x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_plus_a_squared | [
"Primitive of Reciprocal of x cubed by x squared plus a squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x cubed by x squared plus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9263 | Primitive of Reciprocal of x squared plus a squared squared | :$\ds \int \frac {\d x} {\paren {x^2 + a^2}^2} = \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 + a^2}^2}
| r = \int \frac {a \sec^2 \theta... | :$\ds \int \frac {\d x} {\paren {x^2 + a^2}^2} = \frac x {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^3} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 + a^2}^2}
| r = \int \frac {a \sec^2 \t... | Primitive of Reciprocal of x squared plus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Derivative of Tangent Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Square of Cosine Function",
"Tangent Half-Angle Substitution for Sine"
] |
proofwiki-9264 | Primitive of x over x squared plus a squared squared | :$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^2} = -\frac 1 {2 \paren {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^2}
| r = \int \frac {\d z} {2 z^2}
| c = Integration by Sub... | :$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^2} = -\frac 1 {2 \paren {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^2}
| r = \int \frac {\d z} {2 z^2}
| c = [[Integration ... | Primitive of x over x squared plus a squared squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_plus_a_squared_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9265 | Primitive of x squared over x squared plus a squared squared | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {a^2 \tan^2... | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c =
}}
{{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 {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {a^2 \t... | Primitive of x squared over x squared plus a squared squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared/Proof_1 | [
"Primitive of x squared over x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Derivative of Tangent Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Square of Sine Function",
"Tangent Half-Angle Substitution for Sine"
] |
proofwiki-9266 | Primitive of x squared over x squared plus a squared squared | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}
| c = Linear Combination of Primitives
}}
{{e... | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2} = \frac {-x} {2 \paren {x^2 + a^2} } + \frac 1 {2 a} \arctan \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x^2 + a^2 - a^2} {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 + a^2} {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {\d x} {\paren {x^2 + a^2}^2}
| c = [[Linear Combination of Primitives]]
}}... | Primitive of x squared over x squared plus a squared squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_plus_a_squared_squared/Proof_2 | [
"Primitive of x squared over x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared plus a squared squared"
] |
proofwiki-9267 | Primitive of x cubed over x squared plus a squared squared | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {\paren {z - a^2} } {z^2} \frac {\d z} 2
... | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {\paren {z - a^2} } {z^2} \frac {\d z} ... | Primitive of x cubed over x squared plus a squared squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared/Proof_1 | [
"Primitive of x cubed over x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal"
] |
proofwiki-9268 | Primitive of x cubed over x squared plus a squared squared | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x \paren {x^2 + a^2 - a^2} } {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2} } {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}
| c = Linear Combi... | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2}
| r = \int \frac {x \paren {x^2 + a^2 - a^2} } {\paren {x^2 + a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 + a^2} } {\paren {x^2 + a^2}^2} \rd x - a^2 \int \frac {x \rd x} {\paren {x^2 + a^2}^2}
| c = [[Linear Com... | Primitive of x cubed over x squared plus a squared squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_plus_a_squared_squared/Proof_2 | [
"Primitive of x cubed over x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over x squared plus a squared",
"Primitive of x over x squared minus a squared squared"
] |
proofwiki-9269 | Primitive of Reciprocal of x by x squared plus a squared squared | :$\ds \int \frac {\d x} {x \paren {x^2 + a^2}^2} = \frac 1 {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2}^2}
| r = \int \paren {\frac 1 {a^4 x} - \frac x {a^4 \paren {x^2 + a^2} } - \frac x {a^2 \paren {x^2 + a^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}... | :$\ds \int \frac {\d x} {x \paren {x^2 + a^2}^2} = \frac 1 {2 a^2 \paren {x^2 + a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2}^2}
| r = \int \paren {\frac 1 {a^4 x} - \frac x {a^4 \paren {x^2 + a^2} } - \frac x {a^2 \paren {x^2 + a^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x by x squared plus a squared squared/Partial Fraction Expansion|Partial Fraction E... | Primitive of Reciprocal of x by x squared plus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared_squared | [
"Primitive of Reciprocal of x by x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x by x squared plus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Primitive of x over x squared plus a squared squared"
] |
proofwiki-9270 | Primitive of Reciprocal of x squared by x squared plus a squared squared | :$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2} = -\frac 1 {a^4 x} - \frac x {2 a^4 \paren {x^2 + a^2} } - \frac 3 {2 a^5} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} - \frac 1 {a^4 \paren {x^2 + a^2} } - \frac 1 {a^2 \paren {x^2 + a^2}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^4} \int \frac {\d x} {x^2} - \frac 1 {a^4} \int \frac {... | :$\ds \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2} = -\frac 1 {a^4 x} - \frac x {2 a^4 \paren {x^2 + a^2} } - \frac 3 {2 a^5} \arctan \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 + a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} - \frac 1 {a^4 \paren {x^2 + a^2} } - \frac 1 {a^2 \paren {x^2 + a^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by x squared plus a squared squared/Partial Fraction Expansion|Partia... | Primitive of Reciprocal of x squared by x squared plus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_plus_a_squared_squared | [
"Primitive of Reciprocal of x squared by x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x squared by x squared plus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared plus a squared squared"
] |
proofwiki-9271 | Primitive of Reciprocal of x cubed by x squared plus a squared squared | :$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2} = -\frac 1 {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 + a^2} } - \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2}
| r = \int \paren {-\frac 2 {a^6 x} + \frac 1 {a^4 x^3} + \frac {2 x} {a^6 \paren {x^2 + a^2} } + \frac x {a^4 \paren {x^2 + a^2}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = -\frac 2 {a^6} \int \frac {\d x} x + \frac ... | :$\ds \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2} = -\frac 1 {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 + a^2} } - \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 + a^2} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 + a^2}^2}
| r = \int \paren {-\frac 2 {a^6 x} + \frac 1 {a^4 x^3} + \frac {2 x} {a^6 \paren {x^2 + a^2} } + \frac x {a^4 \paren {x^2 + a^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x cubed by x squared plus a squared squared/Partial Frac... | Primitive of Reciprocal of x cubed by x squared plus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_plus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_plus_a_squared_squared | [
"Primitive of Reciprocal of x cubed by x squared plus a squared squared",
"Primitives involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x cubed by x squared plus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Power",
"Primitive of x over x squared plus a squared",
"Primitive of x over x squared plus a squared squared"
] |
proofwiki-9272 | Primitive of Reciprocal of Power of x squared plus a squared | :$\ds \int \frac {\d x} {\paren {x^2 + a^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }$ | 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 = Powe... | :$\ds \int \frac {\d x} {\paren {x^2 + a^2}^n} = \frac x {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 + a^2}^{n - 1} }$ | 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 ... | Primitive of Reciprocal of Power of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_squared_plus_a_squared | [
"Primitives involving x squared plus a 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-9273 | Primitive of x over Power of x squared plus a squared | :$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 + a^2}^{n - 1} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^n}
| r = \int \frac {\d z} {2 z^n}
| c = Integration... | :$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 + a^2}^{n - 1} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^n}
| r = \int \frac {\d z} {2 z^n}
| c = [[Integ... | Primitive of x over Power of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_x_squared_plus_a_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9274 | Primitive of Reciprocal of x by Power of x squared plus a squared | :$\ds \int \frac {\d x} {x \paren {x^2 + a^2}^n} = \frac 1 {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {x^2 + a^2}^{n - 1} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^n}
| r = \int \frac {\d z} {2 x^2 z^n}
| c = Integrat... | :$\ds \int \frac {\d x} {x \paren {x^2 + a^2}^n} = \frac 1 {2 \paren {n - 1} a^2 \paren {x^2 + a^2}^{n - 1} } + \frac 1 {a^2} \int \frac {\d x} {x \paren {x^2 + a^2}^{n - 1} }$ | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 + a^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 {x^2 + a^2}^n}
| r = \int \frac {\d z} {2 x^2 z^n}
| c = [[In... | Primitive of Reciprocal of x by Power of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_squared_plus_a_squared | [
"Primitives involving x squared plus a 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-9275 | Primitive of Power of x over Power of x squared plus a squared | :$\ds \int \frac {x^m \rd x} {\paren {x^2 + a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^{n - 1} } - a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^n}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^2 + a^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {x^2 + a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 + a^2 - a^2} \rd x} {\paren {x^2 + a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 + a^... | :$\ds \int \frac {x^m \rd x} {\paren {x^2 + a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^{n - 1} } - a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 + a^2}^n}$ | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^2 + a^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {x^2 + a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 + a^2 - a^2} \rd x} {\paren {x^2 + a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 + a^... | Primitive of Power of x over Power of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_x_squared_plus_a_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9276 | Primitive of Reciprocal of Power of x by Power of x squared plus a squared | :$\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} }
| r = \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{n - 1} \paren {x^2 + a^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{\left({n - 1}\right) + 1} }
| c =
}}
{{eq... | :$\ds \int \frac {\d x} {x^m \paren {x^2 + a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 + a^2}^n}$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^2 + a^2}^{n - 1} }
| r = \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{n - 1} \paren {x^2 + a^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2} \rd x} {x^m \paren {x^2 + a^2}^{\left({n - 1}\right) + 1} }
| c =
}}
{{eq... | Primitive of Reciprocal of Power of x by Power of x squared plus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_x_squared_plus_a_squared | [
"Primitives involving x squared plus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9277 | Primitive of x over x squared minus a squared | :$\ds \int \frac {x \rd x} {x^2 - a^2} = \frac 1 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2} }
| r = \int \frac {\d z} {2 z}
| c = Integration by ... | :$\ds \int \frac {x \rd x} {x^2 - a^2} = \frac 1 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2} }
| r = \int \frac {\d z} {2 z}
| c = [[Integrati... | Primitive of x over x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_minus_a_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-9278 | Primitive of x cubed over x squared minus a squared | :$\ds \int \frac {x^3 \rd x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^2 - a^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2} } {x^2 - a^2} \rd x + \int \frac {a^2 x} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int x \rd x + a^2 \int \frac {x \r... | :$\ds \int \frac {x^3 \rd x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {x^2 - a^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2} } {x^2 - a^2} \rd x + \int \frac {a^2 x} {x^2 - a^2} \rd x
| c =
}}
{{eqn | r = \int x \rd x + a^2 \int \frac {x \r... | Primitive of x cubed over x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_minus_a_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Power",
"Primitive of x over x squared minus a squared"
] |
proofwiki-9279 | Primitive of Reciprocal of x by x squared minus a squared | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 - a^2} }
| r = \int \paren {\frac x {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {x \rd x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} x
| c = Linear Combinatio... | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2} } = \frac 1 {2 a^2} \map \ln {\frac {x^2 - a^2} {x^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 - a^2} }
| r = \int \paren {\frac x {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x} } \rd x
| c = [[Primitive of Reciprocal of x by x squared minus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2} \int \fra... | Primitive of Reciprocal of x by x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_minus_a_squared | [
"Primitive of Reciprocal of x by x squared minus a squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x by x squared minus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared minus a squared",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9280 | Primitive of Reciprocal of x squared by x squared minus a squared | :$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2} } = \frac 1 {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 - a^2} }
| r = \int \paren {\frac 1 {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} {x^2 - a^2} - \frac 1 {a^2} \int \frac {\d x} {x^2}
| c = Linear Combin... | :$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2} } = \frac 1 {a^2 x} + \frac 1 {2 a^3} \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 - a^2} }
| r = \int \paren {\frac 1 {a^2 \paren {x^2 - a^2} } - \frac 1 {a^2 x^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by x squared minus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2... | Primitive of Reciprocal of x squared by x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_minus_a_squared | [
"Primitive of Reciprocal of x squared by x squared minus a squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x squared by x squared minus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form"
] |
proofwiki-9281 | Primitive of Reciprocal of x cubed by x squared minus a squared | :$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2} } = \frac 1 {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2 - a^2} {x^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 - a^2} }
| r = \int \paren {\frac {-1} {a^2 x^3} - \frac 1 {a^4 x} + \frac x {a^4 \paren {x^2 - a^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 {... | :$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2} } = \frac 1 {2 a^2 x^2} + \frac 1 {2 a^4} \map \ln {\frac {x^2 - a^2} {x^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 - a^2} }
| r = \int \paren {\frac {-1} {a^2 x^3} - \frac 1 {a^4 x} + \frac x {a^4 \paren {x^2 - a^2} } } \rd x
| c = [[Primitive of Reciprocal of x cubed by x squared minus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn... | Primitive of Reciprocal of x cubed by x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_minus_a_squared | [
"Primitive of Reciprocal of x cubed by x squared minus a squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x cubed by x squared minus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of x over x squared minus a squared",
"Logarithm of Power",
"Difference of Logarithms"
] |
proofwiki-9282 | Primitive of Reciprocal of x squared minus a squared squared | :$\ds \int \frac {\d x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {4 a^3 \paren {x + a} } - \frac 1 {4 a^3 \paren {x - a} } + \frac 1 {4 a^2 \paren {x + a}^2} + \frac 1 {4 a^2 \paren {x - a}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {4 a^3} \int \frac {\d... | :$\ds \int \frac {\d x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {4 a^3} \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {4 a^3 \paren {x + a} } - \frac 1 {4 a^3 \paren {x - a} } + \frac 1 {4 a^2 \paren {x + a}^2} + \frac 1 {4 a^2 \paren {x - a}^2} } \rd x
| c = [[Primitive of Reciprocal of x squared minus a squared squared/Partial F... | Primitive of Reciprocal of x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared_squared | [
"Primitive of Reciprocal of x squared minus a squared squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Function of a x + b",
"Primitive of Reciprocal",
"Difference of Logarithms",
"Sign of Quotient of Factors of Difference of Squares",
"Primitive of Functi... |
proofwiki-9283 | Chapman-Kolmogorov Equation | Let $X$ be a homogeneous Markov chain with $n$-step transition probability matrix:
:$\mathbf P^{\paren n} = \sqbrk { {p_{j k} }^{\paren n} }_{j, k \mathop \in S}$
where:
:${p_{j k} }^{\paren n} = \condprob {X_n = k} {X_0 = j} $ is the $n$-step transition probability.
Then:
:$\mathbf P^{\paren {n + m} } = \mathbf P^{\pa... | We consider the conditional probability on the {{LHS}}:
{{begin-eqn}}
{{eqn | l = \ds {p_{i j} }^{\paren {n + m} }
| r = \map \Pr {X_{m + n} = j \mid X_0 = i}
}}
{{eqn | r = \condprob {\paren {\bigcup_{k \mathop \in S} \sqbrk {X_{n + m} = j, X_n = k} } } {X_0 = i}
}}
{{eqn | r = \sum_{k \mathop \in S} \condprob {... | Let $X$ be a [[Definition:Markov Chain/Homogeneous|homogeneous]] [[Definition:Markov Chain|Markov chain]] with $n$-step [[Definition:Transition Probability Matrix|transition probability matrix]]:
:$\mathbf P^{\paren n} = \sqbrk { {p_{j k} }^{\paren n} }_{j, k \mathop \in S}$
where:
:${p_{j k} }^{\paren n} = \condprob {... | We consider the [[Definition:Conditional Probability|conditional probability]] on the {{LHS}}:
{{begin-eqn}}
{{eqn | l = \ds {p_{i j} }^{\paren {n + m} }
| r = \map \Pr {X_{m + n} = j \mid X_0 = i}
}}
{{eqn | r = \condprob {\paren {\bigcup_{k \mathop \in S} \sqbrk {X_{n + m} = j, X_n = k} } } {X_0 = i}
}}
{{eqn ... | Chapman-Kolmogorov Equation | https://proofwiki.org/wiki/Chapman-Kolmogorov_Equation | https://proofwiki.org/wiki/Chapman-Kolmogorov_Equation | [
"Markov Chains"
] | [
"Definition:Markov Chain/Homogeneous",
"Definition:Markov Chain",
"Definition:Transition Probability Matrix",
"Definition:Markov Chain/Transition Probability"
] | [
"Definition:Conditional Probability",
"Chain Rule for Probability",
"Markov Property",
"Transition Probabilities of Homogeneous Markov Chain",
"Category:Markov Chains"
] |
proofwiki-9284 | Primitive of x over x squared minus a squared squared | :$\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^2} = \frac {-1} {2 \paren {x^2 - a^2} } + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^2}
| r = \int \frac {\d z} {2 z^2}
| c = Integration by Sub... | :$\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^2} = \frac {-1} {2 \paren {x^2 - a^2} } + C$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^2}
| r = \int \frac {\d z} {2 z^2}
| c = [[Integration ... | Primitive of x over x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_over_x_squared_minus_a_squared_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-9285 | Primitive of x squared over x squared minus a squared squared | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 \paren {x^2 - a^2} } + \frac 1 {4 a} \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2}
| r = \int \frac {x^2 - a^2 + a^2} {\paren {x^2 - a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {\paren {x^2 - a^2}^2} \rd x + a^2 \int \frac {\d x} {\paren {x^2 - a^2}^2}
| c = Linear Combination of Primitives
}}
{{e... | :$\ds \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2} = \frac {-x} {2 \paren {x^2 - a^2} } + \frac 1 {4 a} \map \ln {\frac {x - a} {x + a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^2 \rd x} {\paren {x^2 - a^2}^2}
| r = \int \frac {x^2 - a^2 + a^2} {\paren {x^2 - a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x^2 - a^2} {\paren {x^2 - a^2}^2} \rd x + a^2 \int \frac {\d x} {\paren {x^2 - a^2}^2}
| c = [[Linear Combination of Primitives]]
}}... | Primitive of x squared over x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_squared_over_x_squared_minus_a_squared_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form",
"Primitive of Reciprocal of x squared minus a squared squared"
] |
proofwiki-9286 | Primitive of x cubed over x squared minus a squared squared | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2} = \frac {-a^2} {2 \paren {x^2 - a^2} } + \frac 1 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {\paren {x^2 - a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2} } {\paren {x^2 - a^2}^2} \rd x + a^2 \int \frac {x \rd x} {\paren {x^2 - a^2}^2}
| c = Linear Combi... | :$\ds \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2} = \frac {-a^2} {2 \paren {x^2 - a^2} } + \frac 1 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2}
| r = \int \frac {x \paren {x^2 - a^2 + a^2} } {\paren {x^2 - a^2}^2} \rd x
| c =
}}
{{eqn | r = \int \frac {x \paren {x^2 - a^2} } {\paren {x^2 - a^2}^2} \rd x + a^2 \int \frac {x \rd x} {\paren {x^2 - a^2}^2}
| c = [[Linear Com... | Primitive of x cubed over x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_x_cubed_over_x_squared_minus_a_squared_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite",
"Primitive of x over x squared minus a squared",
"Primitive of x over x squared minus a squared squared"
] |
proofwiki-9287 | Primitive of Reciprocal of x by x squared minus a squared squared | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x} + \frac {-x} {a^4 \paren {x^2 - a^2} } + \frac x {a^2 \paren {x^2 - a^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... | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2}^2} = \frac {-1} {2 a^2 \paren {x^2 - a^2} } + \frac 1 {2 a^4} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x} + \frac {-x} {a^4 \paren {x^2 - a^2} } + \frac x {a^2 \paren {x^2 - a^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x by x squared minus a squared squared/Partial Fraction Expansion|Partial Fraction Ex... | Primitive of Reciprocal of x by x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_minus_a_squared_squared | [
"Primitive of Reciprocal of x by x squared minus a squared squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x by x squared minus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared minus a squared",
"Primitive of x over x squared minus a squared squared",
"Logarithm of Power",
"Differ... |
proofwiki-9288 | Primitive of Reciprocal of x squared by x squared minus a squared squared | :$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \paren {x^2 - a^2} } + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {x + a} } - \frac 3 {4 a^5 \paren {x - a} } + \frac 1 {4 a^4 \paren {x + a}^2} + \frac 1 {4 a^4 \paren {x - a}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac ... | :$\ds \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2} = \frac {-1} {a^4 x} - \frac x {2 a^4 \paren {x^2 - a^2} } + \frac 3 {4 a^5} \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^2} + \frac 3 {4 a^5 \paren {x + a} } - \frac 3 {4 a^5 \paren {x - a} } + \frac 1 {4 a^4 \paren {x + a}^2} + \frac 1 {4 a^4 \paren {x - a}^2} } \rd x
| c = [[Primitive of Reciprocal of x squared by x squa... | Primitive of Reciprocal of x squared by x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_x_squared_minus_a_squared_squared | [
"Primitive of Reciprocal of x squared by x squared minus a squared squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x squared by x squared minus a 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-9289 | Primitive of Reciprocal of x cubed by x squared minus a squared squared | :$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^3} + \frac 2 {a^6 x} - \frac {2 x} {a^6 \paren {x^2 - a^2} } + \frac x {a^4 \paren {x^2 - a^2}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^4} \int \frac {\d x} {x^3} + \frac 2 {... | :$\ds \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2} = \frac {-1} {2 a^4 x^2} - \frac 1 {2 a^4 \paren {x^2 - a^2} } + \frac 1 {a^6} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {x^2 - a^2}^2}
| r = \int \paren {\frac 1 {a^4 x^3} + \frac 2 {a^6 x} - \frac {2 x} {a^6 \paren {x^2 - a^2} } + \frac x {a^4 \paren {x^2 - a^2}^2} } \rd x
| c = [[Primitive of Reciprocal of x cubed by x squared minus a squared squared/Partial Fraction ... | Primitive of Reciprocal of x cubed by x squared minus a squared squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_minus_a_squared_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_x_squared_minus_a_squared_squared | [
"Primitive of Reciprocal of x cubed by x squared minus a squared squared",
"Primitives involving x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x cubed by x squared minus a squared squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of x over x squared minus a squared",
"Primitive of x over x squared minus a squared squared",
"... |
proofwiki-9290 | Primitive of Reciprocal of Power of x squared minus a squared | :$\ds \int \frac {\d x} {\paren {x^2 - a^2}^n} = \frac {-x} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 - a^2}^{n - 1} }$
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... | :$\ds \int \frac {\d x} {\paren {x^2 - a^2}^n} = \frac {-x} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac {2 n - 3} {\paren {2 n - 2} a^2} \int \frac {\d x} {\paren {x^2 - a^2}^{n - 1} }$
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 x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_squared_minus_a_squared | [
"Primitives involving x squared minus a 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-9291 | Primitive of x over Power of x squared minus a squared | :$\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^n}
| r = \int \frac {\d z} {2 z^n}
| c = Integration... | :$\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^n}
| r = \int \frac {\d z} {2 z^n}
| c = [[Integ... | Primitive of x over Power of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_over_Power_of_x_squared_minus_a_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Primitive of Power"
] |
proofwiki-9292 | Primitive of Reciprocal of x by Power of x squared minus a squared | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^n}
| r = \int \frac {\d z} {2 x^2 z^n}
| c = Integrat... | :$\ds \int \frac {\d x} {x \paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} a^2 \paren {x^2 - a^2}^{n - 1} } - \frac 1 {a^2} \int \frac {\d x} {x \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | Let:
{{begin-eqn}}
{{eqn | l = z
| r = x^2 - a^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 {x^2 - a^2}^n}
| r = \int \frac {\d z} {2 x^2 z^n}
| c = [[In... | Primitive of Reciprocal of x by Power of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Power_of_x_squared_minus_a_squared | [
"Primitives involving x squared minus a 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-9293 | Primitive of Power of x over Power of x squared minus a squared | :$\ds \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {x^2 - a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 - a^2 + a^2} \rd x} {\paren {x^2 - a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 - a^... | :$\ds \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n} = \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^{n - 1} } + a^2 \int \frac {x^{m - 2} \rd x} {\paren {x^2 - a^2}^n}$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^2 - a^2}^n}
| r = \int \frac {x^{m - 2} \paren {x^2} \rd x} {\paren {x^2 - a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 - a^2 + a^2} \rd x} {\paren {x^2 - a^2}^n}
| c =
}}
{{eqn | r = \int \frac {x^{m - 2} \paren {x^2 - a^... | Primitive of Power of x over Power of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Power_of_x_over_Power_of_x_squared_minus_a_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9294 | Primitive of Reciprocal of Power of x by Power of x squared minus a squared | :$\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }
| r = \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{n - 1} \paren {x^2 - a^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{\paren {n - 1} + 1} }
| c =
}}
{{eqn | r ... | :$\ds \int \frac {\d x} {x^m \paren {x^2 - a^2}^n} = \frac 1 {a^2} \int \frac {\d x} {x^{m - 2} \paren {x^2 - a^2}^n} - \frac 1 {a^2} \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^2 - a^2}^{n - 1} }
| r = \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{n - 1} \paren {x^2 - a^2} }
| c =
}}
{{eqn | r = \int \frac {\paren {x^2 - a^2} \rd x} {x^m \paren {x^2 - a^2}^{\paren {n - 1} + 1} }
| c =
}}
{{eqn | r ... | Primitive of Reciprocal of Power of x by Power of x squared minus a squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_x_squared_minus_a_squared | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Power_of_x_by_Power_of_x_squared_minus_a_squared | [
"Primitives involving x squared minus a squared"
] | [] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-9295 | Sign of Quotient of Factors of Difference of Squares | Let $a, b \in \R$ such that $a \ne b$.
Then
:$\map \sgn {a^2 - b^2} = \map \sgn {\dfrac {a + b} {a - b} } = \map \sgn {\dfrac {a - b} {a + b} }$
where $\sgn$ denotes the signum of a real number. | {{begin-eqn}}
{{eqn | l = \map \sgn {\frac {a - b} {a + b} }
| r = \map \sgn {a - b} \frac 1 {\map \sgn {a + b} }
| c = Signum Function is Completely Multiplicative
}}
{{eqn | r = \map \sgn {a - b} \map \sgn {a + b}
| c = Signum Function of Reciprocal
}}
{{eqn | r = \map \sgn {\paren {a - b} \paren {a... | Let $a, b \in \R$ such that $a \ne b$.
Then
:$\map \sgn {a^2 - b^2} = \map \sgn {\dfrac {a + b} {a - b} } = \map \sgn {\dfrac {a - b} {a + b} }$
where $\sgn$ denotes the [[Definition:Signum Function|signum]] of a [[Definition:Real Number|real number]]. | {{begin-eqn}}
{{eqn | l = \map \sgn {\frac {a - b} {a + b} }
| r = \map \sgn {a - b} \frac 1 {\map \sgn {a + b} }
| c = [[Signum Function is Completely Multiplicative]]
}}
{{eqn | r = \map \sgn {a - b} \map \sgn {a + b}
| c = [[Signum Function of Reciprocal]]
}}
{{eqn | r = \map \sgn {\paren {a - b} \... | Sign of Quotient of Factors of Difference of Squares | https://proofwiki.org/wiki/Sign_of_Quotient_of_Factors_of_Difference_of_Squares | https://proofwiki.org/wiki/Sign_of_Quotient_of_Factors_of_Difference_of_Squares | [
"Real Analysis",
"Signum Function"
] | [
"Definition:Signum Function",
"Definition:Real Number"
] | [
"Signum Function is Completely Multiplicative",
"Signum Function of Reciprocal",
"Signum Function is Completely Multiplicative",
"Difference of Two Squares",
"Difference of Two Squares",
"Signum Function is Completely Multiplicative",
"Signum Function of Reciprocal",
"Signum Function is Completely Mul... |
proofwiki-9296 | Signum Function is Completely Multiplicative | The signum function on the set of real numbers is a completely multiplicative function:
:$\forall x, y \in \R: \map \sgn {x y} = \map \sgn x \map \sgn y$ | Let $x = 0$ or $y = 0$.
Then:
{{begin-eqn}}
{{eqn | l = x y
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \map \sgn {x y}
| r = 0
| c =
}}
{{end-eqn}}
and either $\map \sgn x = 0$ or $\map \sgn y = 0$ and so:
{{begin-eqn}}
{{eqn | l = \map \sgn x \map \sgn y
| r = 0
... | The [[Definition:Signum Function|signum function]] on the set of [[Definition:Real Number|real numbers]] is a [[Definition:Completely Multiplicative Function|completely multiplicative function]]:
:$\forall x, y \in \R: \map \sgn {x y} = \map \sgn x \map \sgn y$ | Let $x = 0$ or $y = 0$.
Then:
{{begin-eqn}}
{{eqn | l = x y
| r = 0
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \map \sgn {x y}
| r = 0
| c =
}}
{{end-eqn}}
and either $\map \sgn x = 0$ or $\map \sgn y = 0$ and so:
{{begin-eqn}}
{{eqn | l = \map \sgn x \map \sgn y
| r = 0... | Signum Function is Completely Multiplicative | https://proofwiki.org/wiki/Signum_Function_is_Completely_Multiplicative | https://proofwiki.org/wiki/Signum_Function_is_Completely_Multiplicative | [
"Signum Function",
"Completely Multiplicative Functions"
] | [
"Definition:Signum Function",
"Definition:Real Number",
"Definition:Completely Multiplicative Function"
] | [
"Category:Signum Function",
"Category:Completely Multiplicative Functions"
] |
proofwiki-9297 | Signum Function is Quotient of Number with Absolute Value | Let $x \in \R_{\ne 0}$ be a non-zero real number.
Then:
:$\map \sgn x = \dfrac x {\size x} = \dfrac {\size x} x$
where:
:$\map \sgn x$ denotes the signum function of $x$
:$\size x$ denotes the absolute value of $x$. | Let $x \in \R_{\ne 0}$.
Then either $x > 0$ or $x < 0$.
Let $x > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \frac x {\size x}
| r = \frac x x
| c = {{Defof|Absolute Value}}, as $x > 0$
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = \map \sgn x
| c = {{Defof|Signum Function}}, as $x > 0$
}}
{{end-eqn}}
Similarl... | Let $x \in \R_{\ne 0}$ be a [[Definition:Zero (Number)|non-zero]] [[Definition:Real Number|real number]].
Then:
:$\map \sgn x = \dfrac x {\size x} = \dfrac {\size x} x$
where:
:$\map \sgn x$ denotes the [[Definition:Signum Function|signum function]] of $x$
:$\size x$ denotes the [[Definition:Absolute Value|absolute va... | Let $x \in \R_{\ne 0}$.
Then either $x > 0$ or $x < 0$.
Let $x > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \frac x {\size x}
| r = \frac x x
| c = {{Defof|Absolute Value}}, as $x > 0$
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = \map \sgn x
| c = {{Defof|Signum Function}}, as $x > 0$
}}
{{end-eqn}}
Si... | Signum Function is Quotient of Number with Absolute Value | https://proofwiki.org/wiki/Signum_Function_is_Quotient_of_Number_with_Absolute_Value | https://proofwiki.org/wiki/Signum_Function_is_Quotient_of_Number_with_Absolute_Value | [
"Signum Function",
"Absolute Value Function"
] | [
"Definition:Zero (Number)",
"Definition:Real Number",
"Definition:Signum Function",
"Definition:Absolute Value"
] | [
"Category:Signum Function",
"Category:Absolute Value Function"
] |
proofwiki-9298 | Completely Multiplicative Function of Quotient | Let $f: \R \to \R$ be a completely multiplicative function.
Then:
:$\forall x, y \in \R, y \ne 0: \map f {\dfrac x y} = \dfrac {\map f x} {\map f y}$
whenever $\map f y \ne 0$. | Let $z = \dfrac x y$.
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \dfrac x y
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y z
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x
| r = \map f y \map f z
| c = {{Defof|Completely Multiplicative Function}}
}}
{{eqn | ll= \leadsto
... | Let $f: \R \to \R$ be a [[Definition:Completely Multiplicative Function|completely multiplicative function]].
Then:
:$\forall x, y \in \R, y \ne 0: \map f {\dfrac x y} = \dfrac {\map f x} {\map f y}$
whenever $\map f y \ne 0$. | Let $z = \dfrac x y$.
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \dfrac x y
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y z
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x
| r = \map f y \map f z
| c = {{Defof|Completely Multiplicative Function}}
}}
{{eqn | ll= \leadsto
... | Completely Multiplicative Function of Quotient | https://proofwiki.org/wiki/Completely_Multiplicative_Function_of_Quotient | https://proofwiki.org/wiki/Completely_Multiplicative_Function_of_Quotient | [
"Multiplicative Functions"
] | [
"Definition:Completely Multiplicative Function"
] | [
"Category:Multiplicative Functions"
] |
proofwiki-9299 | Signum Function of Reciprocal | Let $x \in \R$ such that $x \ne 0$.
Then:
:$\map \sgn x = \map \sgn {\dfrac 1 x}$
where $\map \sgn x$ denotes the signum of $x$. | {{begin-eqn}}
{{eqn | l = \map \sgn x
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = >
| r = 0
| c = {{Defof|Signum Function}}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| o = >
| r = 0
| c = Reciprocal of Strictly Positive Real Number is Stri... | Let $x \in \R$ such that $x \ne 0$.
Then:
:$\map \sgn x = \map \sgn {\dfrac 1 x}$
where $\map \sgn x$ denotes the [[Definition:Signum Function|signum]] of $x$. | {{begin-eqn}}
{{eqn | l = \map \sgn x
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = >
| r = 0
| c = {{Defof|Signum Function}}
}}
{{eqn | ll= \leadstoandfrom
| l = \frac 1 x
| o = >
| r = 0
| c = [[Reciprocal of Strictly Positive Real Number is St... | Signum Function of Reciprocal | https://proofwiki.org/wiki/Signum_Function_of_Reciprocal | https://proofwiki.org/wiki/Signum_Function_of_Reciprocal | [
"Signum Function"
] | [
"Definition:Signum Function"
] | [
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Category:Signum Function"
] |
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