nyuuzyou/wikis huggingface dataset:
Primitive of Power of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x + C$
for $n \ne -1$.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\sinh^n a x$
* Primitive of $\cosh^n a x$
* Primitive of $\tanh^n a x$
* Primitive of $\coth^n a x$
* Primitive of $\sech^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.645$
* : Back endpapers: A Brief Table of Integrals: $134$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.7.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$
== Proof ==
== Also see ==
* Primitive of $\sech^n a x \tanh a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.639$
* : Back endpapers: A Brief Table of Integrals: $136$.
Category:Primitives involving Hyperbolic Cosecant Function
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Power of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x$
for $n \ne 0$.
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
We note that if $n = 0$, then $\dfrac {n - 1} n$ is undefined, rendering the derivation invalid.
== Also see ==
* Primitive of $\sinh^n a x$
* Primitive of $\tanh^n a x$
* Primitive of $\coth^n a x$
* Primitive of $\sech^n a x$
* Primitive of $\csch^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.586$
* : Back endpapers: A Brief Table of Integrals: $118$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.13.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
== Proof ==
== Also see ==
* Primitive of $\tanh a x$ for the case where $n = -1$.
* Primitive of $\sinh^n a x \cosh a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.593$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Power of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^n a x$
* Primitive of $\cosh^n a x$
* Primitive of $\tanh^n a x$
* Primitive of $\sech^n a x$
* Primitive of $\csch^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.625$
* : Back endpapers: A Brief Table of Integrals: $128$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.8.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$
== Proof ==
== Also see ==
* Primitive of $\tanh^n a x \sech^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.618$
Category:Primitives involving Hyperbolic Cotangent Function
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of Power of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$
for $n \ne -1$.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\sinh^n a x$
* Primitive of $\cosh^n a x$
* Primitive of $\tanh^n a x$
* Primitive of $\coth^n a x$
* Primitive of $\csch^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.635$
* : Back endpapers: A Brief Table of Integrals: $133$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\sech a x$: $17.32.7.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$
for $n \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\csch^n a x \coth a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.629$
* : Back endpapers: A Brief Table of Integrals: $135$.
Category:Primitives involving Hyperbolic Secant Function
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Power of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x$
for $n \ne 0$.
== Proof ==
With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
We note that if $n = 0$, then $\dfrac {n - 1} n$ is undefined, rendering the derivation invalid.
== Also see ==
* Primitive of $\cosh^n a x$
* Primitive of $\tanh^n a x$
* Primitive of $\coth^n a x$
* Primitive of $\sech^n a x$
* Primitive of $\csch^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.558$
* : Back endpapers: A Brief Table of Integrals: $117$.
Category:Primitives involving Hyperbolic Sine Function
Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
== Proof ==
== Also see ==
* Primitive of $\coth a x$ for the case where $n = -1$.
* Primitive of $\cosh^n a x \sinh a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.592$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Power of Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \tanh^n a x \rd x = \frac {-\tanh^{n - 1} a x} {a \paren {n - 1} } + \int \tanh^{n - 2} a x \rd x + C$
== Proof ==
== Also see ==
* Primitive of $\sinh^n a x$
* Primitive of $\cosh^n a x$
* Primitive of $\coth^n a x$
* Primitive of $\sech^n a x$
* Primitive of $\csch^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.614$
* : Back endpapers: A Brief Table of Integrals: $127$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.8.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int \tanh^n a x \sech^2 a x \rd x = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C$
== Proof ==
== Also see ==
* Primitive of $\coth^n a x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.607$
Category:Primitives involving Hyperbolic Tangent Function
Category:Primitives involving Hyperbolic Secant Function
Primitive of Power of Logarithm of x

== Theorem ==
:$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\ln x$: $14.535$
* : Appendix: Table $3$: Reduction formulae
* : Appendix: Table $3$: Reduction formulae
* : Appendix $8$: Integrals: Reduction Formulae
Category:Primitives involving Logarithm Function
Primitive of Power of Logarithm of x over x

== Theorem ==
:$\ds \int \frac {\ln^n x} x \rd x = \frac {\ln^{n + 1} x} {n + 1} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $\ln x$: $14.531$
Category:Primitives involving Logarithm Function
Primitive of Power of Root of 2 a x minus x squared

== Theorem ==
:$\ds \int \paren {\sqrt {2 a x - x^2} }^n \rd x = \frac {\paren {x - a} \paren {\sqrt {2 a x - x^2} }^n} {n + 1} + \frac {n a^2} {n + 1} \int \paren {\sqrt {2 a x - x^2} }^{n - 2} \rd x$
== Proof ==
Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
and we have:
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $49$.
Category:Primitives of Roots of Quadratic Functions
Primitive of Power of Root of a squared minus x squared

== Theorem ==
:$\ds \int \paren {\sqrt {a^2 - x^2} }^n \rd x = \dfrac {x \paren {\sqrt {a^2 - x^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {a^2 - x^2} }^{n - 2} \rd x$
for $n \ne -1$.
== Proof ==
Let:
Also:
Thus:
== Also see ==
For $n = -1$, use Primitive of Reciprocal of Root of a squared minus x squared
Category:Primitive of Power of Root of a squared minus x squared
Category:Primitives involving Root of a squared minus x squared
Primitive of Power of Root of a x + b

== Theorem ==
:$\ds \int \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C$
for $m \ne -2$.
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.99$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.16.$
Category:Primitive of Power of Root of a x + b
Category:Primitives involving Root of a x + b
Primitive of Power of Root of a x + b/Also presented as

== Primitive of Power of $\sqrt {a x + b}$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int \paren {\sqrt {a x + b} }^n \rd x = \frac 2 a \frac {\paren {\sqrt {a x + b} }^{n + 2} } {n + 2} + C$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $11$.
Category:Primitive of Power of Root of a x + b
Primitive of Power of Root of a x + b over x

== Theorem ==
:$\ds \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x = \frac {2 \paren {\sqrt {a x + b} }^m } m + b \int \frac {\paren {\sqrt {a x + b} }^{m - 2} } x \rd x$
== Proof ==
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 $m := -1$:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.102$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.19.$
Category:Primitives involving Root of a x + b
Primitive of Power of Root of a x + b over x squared

== Theorem ==
:$\ds \int \frac {\paren {\sqrt {a x + b} }^m} {x^2} \rd x = -\frac {\paren {\sqrt {a x + b} }^{m + 2} } {b x} + \frac {m a} {2 b} \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x$
== Proof ==
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 $m := -2$:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.103$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.20.$
Category:Primitives involving Root of a x + b
Primitive of Power of Root of x squared minus a squared

== Theorem ==
:$\ds \int \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {x \paren {\sqrt {x^2 - a^2} }^n} {n + 1} - \dfrac {n a^2} {n + 1} \int \paren {\sqrt {x^2 - a^2} }^{n - 2} \rd x$
for $n \ne -1$
== Proof ==
Let:
Also:
Thus:
== Also see ==
For $n = -1$, use Primitive of Reciprocal of Root of x squared minus a squared
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $38$.
Category:Primitive of Power of Root of x squared minus a squared
Category:Primitives involving Root of x squared minus a squared
Primitive of Power of Secant of a x

== Theorem ==
:$\ds \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$
where $n \ne 1$.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u}{\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\sec a x$ for the case where $n = 1$
* Primitive of $\sin^n a x$
* Primitive of $\cos^n a x$
* Primitive of $\tan^n a x$
* Primitive of $\cot^n a x$
* Primitive of $\csc^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.460$
* : Back endpapers: A Brief Table of Integrals: $92$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(22)$ Integrals Involving $\sec a x$: $17.22.10.$
Category:Primitives involving Secant Function
Primitive of Power of Secant of a x by Tangent of a x

== Theorem ==
:$\ds \int \sec^n a x \tan a x \rd x = \frac {\sec^n a x} {n a} + C$
for $n \ne 0$.
== Proof ==
== Also see ==
* Primitive of $\csc^n a x \cot a x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.454$
* : Back endpapers: A Brief Table of Integrals: $94$.
Category:Primitives involving Secant Function
Category:Primitives involving Tangent Function
Primitive of Power of Sine of a x by Cosine of a x

== Theorem ==
:$\ds \int \sin^n a x \cos a x \rd x = \frac {\sin^{n + 1} a x} {\paren {n + 1} a} + C$
for $n \ne -1$.
== Proof ==
== Also see ==
* Primitive of $\cos^n a x \sin a x$
* For $n = -1$, use Primitive of $\dfrac {\cos a x} {\sin a x}$.
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.401$
* : Front endpapers: A Brief Table of Integrals: $64$.
Category:Primitives involving Sine Function and Cosine Function
Primitive of Power of Sine of a x by Power of Cosine of a x

== Theorem ==
=== Reduction of Power of Sine ===
=== Reduction of Power of Cosine ===
== Examples ==
== Also see ==
* Primitive of $\tan^n a x$ for $n = -m$.
* Primitive of $\cot^n a x$ for $m = -n$.
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Category:Primitives involving Sine Function and Cosine Function
Primitive of Power of Sine of a x by Power of Cosine of a x/Examples

== Examples of Use of Primitive of $\sin^m x \cos^n x$ ==
=== Primitive of $\cos^3 x \sin^4 x$ ===
=== Primitive of $\sin^2 x \cos^3 x$ ===
=== Primitive of $\cos^2 x \sin^4 x$ ===
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x by Power of Cosine of a x/Examples/cos 3rd x sin 4th x

== Example of Use of Primitive of $\sin^m x \cos^n x$ ==
:$\ds \int \cos^3 x \sin^4 x \rd x = \dfrac {\sin^5 x} 5 - \dfrac {\sin^7 x} 7 + C$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Powers of cos and sine: Example 1.
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x by Power of Cosine of a x/Examples/cos squared x sin 4th x

== Example of Use of Primitive of $\sin^m x \cos^n x$ ==
:$\ds \int \cos^2 x \sin^4 x \rd x = \dfrac 1 {32} \paren {2 x - \dfrac {\sin 2 x} 2 - \dfrac {2 \sin 4 x} 4 + \dfrac {\sin 6 x} 6} + C$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Integration: Algebraic Integration: Powers of cos and sine: Example
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x by Power of Cosine of a x/Examples/sin squared x cos cubed x

== Example of Use of Primitive of $\sin^m x \cos^n x$ ==
:$\ds \int \sin^2 x \cos^3 x \rd x = \dfrac {\sin^3 x} 3 - \dfrac {\sin^5 x} 5 + C$
== Proof ==
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $11$.
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine

== Theorem ==
:$\ds \int \sin^m a x \cos^n a x \rd x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \paren {m + n} } + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \rd x + C$
for $n \ne -m$.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
Hence after rearranging:
== Also see ==
* Primitive of $\sin^m a x \cos^n a x$ : Reduction of Power of Sine
* Primitive of $\tan^n a x$ for $n = -m$.
* Primitive of $\cot^n a x$ for $m = -n$.
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.425$
* : Front endpapers: A Brief Table of Integrals: $69$.
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Sine

== Theorem ==
:$\ds \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$
for $n \ne -m$.
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
Hence after rearranging:
== Also see ==
* Primitive of $\sin^m a x \cos^n a x$ : Reduction of Power of Cosine
* Primitive of $\tan^n a x$ for $n = -m$.
* Primitive of $\cot^n a x$ for $m = -n$.
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.425$
* : Front endpapers: A Brief Table of Integrals: $68$.
Category:Primitive of Power of Sine of a x by Power of Cosine of a x
Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Both Powers

== Theorem ==
:$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m - 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\sin^{m - 2} a x} {\cos^{n - 2} a x} \rd x + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
Then:
== Also see ==
* Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$: Reduction of Both Powers
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.426$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine

== Theorem ==
:$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C$
== Proof ==
Hence, rearranging:
== Also see ==
* Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$ : Reduction of $\sin^n$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.426$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine

== Theorem ==
:$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
Thus:
and so:
Then let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.426$
Category:Primitives involving Sine Function and Cosine Function
Primitive of Power of Tangent of a x

== Theorem ==
:$\ds \int \tan^n a x \rd x = \frac {\tan^{n - 1} a x} {\paren {n - 1} a} - \int \tan^{n - 2} a x \rd x$
for $n \ne 1$.
== Proof ==
== Also see ==
* Primitive of $\tan a x$ for $n = 1$
* Primitive of $\sin^n a x$
* Primitive of $\cos^n a x$
* Primitive of $\cot^n a x$
* Primitive of $\sec^n a x$
* Primitive of $\csc^n a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.439$
* : Back endpapers: A Brief Table of Integrals: $86$.
* : Appendix: Table $3$: Reduction formulae
* : Appendix: Table $3$: Reduction formulae
* : $\S 17$: Tables of Special Indefinite Integrals: $(20)$ Integrals Involving $\tan a x$: $17.20.11.$
Category:Primitives involving Tangent Function
Primitive of Power of Tangent of a x by Square of Secant of a x

== Theorem ==
:$\ds \int \tan^n a x \sec^2 a x \rd x = \frac {\tan^{n + 1} a x} {\paren {n + 1} a} + C$
== Proof ==
== Also see ==
* Primitive of $\cot^n a x \csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.432$
Category:Primitives involving Tangent Function
Category:Primitives involving Secant Function
Primitive of Power of a x + b

== Theorem ==
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Also see ==
* Primitive of Reciprocal of $a x + b$ for the case when $n = -1$
== Sources ==
* : Chapter $10$: Integration: $10.4$. Standard integrals: Standard Forms: $\text {(i) (b)}$
* : $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Rational Algebraic Functions: $3.3.14$
* : $\S 14$: Integrals involving $a x + b$: $14.80$
* : Front endpapers: A Brief Table of Integrals: $5$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.14.$
Category:Primitive of Power of a x + b
Category:Primitives involving a x + b
Primitive of Power of a x + b/Proof 2

== Theorem ==
where $n \ne 1$.
== Proof ==
Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
Category:Primitive of Power of a x + b
Primitive of Power of a x + b by Power of p x + q/Decrement of Power

== Theorem ==
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd x$
== Proof ==
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:
In order to make $u \dfrac {\d v} {\d x}$ equal to the integrand, let:
Select $s$ such that $m - s + n + 1 = 0$, and so $s = m + n + 1$:
Other instances of $s$ are left as they are, anticipating that they will cancel out later.
Thus:
== Also defined as ==
This can also be reported as:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^m \paren {p x + q}^{n + 1} } {\paren {m + n + 1} p} + \frac {m \paren {b p - a q} } {\paren {m + n + 1} p} \int \paren {a x + b}^{m - 1} \paren {p x + q}^n \rd x$
by interchanging the roles of $m$ and $n$.
Category: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/Increment of Power

== Theorem ==
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {n + 1} \paren {b p - a q} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - a \paren {m + n + 2} \int \paren {a x + b}^m \paren {p x + q}^{n + 1} \rd x}$
== Proof ==
From Primitive of Power of $a x + b$ by Power of $p x + q$: Decrement of Power:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac {\paren {a x + b}^{m + 1} \paren {p x + q}^n} {\paren {m + n + 1} a} - \frac {n \paren {b p - a q} } {\paren {m + n + 1} a} \int \paren {a x + b}^m \paren {p x + q}^{n - 1} \rd x$
Putting $n + 1$ for $n$, this yields:
== Also defined as ==
This can also be reported as:
:$\ds \int \paren {a x + b}^m \paren {p x + q}^n \rd x = \frac 1 {\paren {m + 1} \paren {a q - b p} } \paren {\paren {a x + b}^{m + 1} \paren {p x + q}^{n + 1} - p \paren {m + n + 2} \int \paren {a x + b}^{m + 1} \paren {p x + q}^n \rd x}$
by interchanging the roles of $m$ and $n$.
Category:Primitive of Power of a x + b by Power of p x + q