nyuuzyou/wikis huggingface dataset:
Primitive of x by Square of Sine of a x

== Theorem ==
:$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$
== 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 $x \cos^2 a x$
* Primitive of $x \tan^2 a x$
* Primitive of $x \cot^2 a x$
* Primitive of $x \sec^2 a x$
* Primitive of $x \csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.348$
Category:Primitives involving Sine Function
Primitive of x by Square of Tangent of a x

== Theorem ==
:$\ds \int x \tan^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} - \frac {x^2} 2 + C$
== 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 $x \sin^2 a x$
* Primitive of $x \cos^2 a x$
* Primitive of $x \cot^2 a x$
* Primitive of $x \sec^2 a x$
* Primitive of $x \csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.437$
Category:Primitives involving Tangent Function
Primitive of x by Tangent of a x

== Theorem ==
:$\ds \int x \tan a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x} ^ 3} 3 + \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
From Power Series Expansion for Tangent Function:
== Also see ==
* Primitive of $x \sin a x$
* Primitive of $x \cos a x$
* Primitive of $x \cot a x$
* Primitive of $x \sec a x$
* Primitive of $x \csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\tan a x$: $14.435$
Category:Primitives involving Tangent Function
Primitive of x cubed by Cosine of a x

== Theorem ==
:$\ds \int x^3 \map \cos {a x} \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \cos a x + \paren {\frac {x^3} a - \frac {6 x} {a^3} } \sin a x + C$
where $C$ is an arbitrary constant.
== 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 $x^3 \sin a x$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.372$
Category:Primitives involving Cosine Function
Primitive of x cubed by Root of a squared minus x squared

== Theorem ==
:$\ds \int x^3 \sqrt {a^2 - x^2} \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^5} 5 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^3} 3 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \sqrt {x^2 + a^2}$
* Primitive of $x^3 \sqrt {x^2 - a^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.247$
Category:Primitives involving Root of a squared minus x squared
Primitive of x cubed by Root of a squared minus x squared cubed

== Theorem ==
:$\ds \int x^3 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {\paren {\sqrt {a^2 - x^2} }^7} 7 - \frac {a^2 \paren {\sqrt {a^2 - x^2} }^5} 5 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x^3 \paren {\sqrt {x^2 - a^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.261$
Category:Primitives involving Root of a squared minus x squared
Primitive of x cubed by Root of x squared minus a squared

== Theorem ==
:$\ds \int x^3 \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^3} 3 + C$
for $\size x \ge a$.
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \sqrt {x^2 + a^2}$
* Primitive of $x^3 \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.219$
Category:Primitives involving Root of x squared minus a squared
Primitive of x cubed by Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int x^3 \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^7} 7 + \frac {a^2 \paren {\sqrt {x^2 - a^2} }^5} 5 + C$
for $\size x \ge a$.
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x^3 \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.233$
Category:Primitives involving Root of x squared minus a squared
Primitive of x cubed by Root of x squared plus a squared

== Theorem ==
:$\ds \int x^3 \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^3} 3 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \sqrt {x^2 - a^2}$
* Primitive of $x^3 \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.192$
Category:Primitives involving Root of x squared plus a squared
Primitive of x cubed by Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x^3 \paren {\sqrt {x^2 - a^2} }^3$
* Primitive of $x^3 \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.206$
Category:Primitives involving Root of x squared plus a squared
Primitive of x cubed by Sine of a x

== Theorem ==
:$\ds \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$
where $C$ is an arbitrary constant.
== 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 $x^3 \cos a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.342$
Category:Primitives involving Sine Function
Primitive of x cubed over Root of a squared minus x squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\sqrt {a^2 - x^2} } = \frac {\paren {\sqrt {a^2 - x^2} }^3} 3 - a^2 \sqrt {a^2 - x^2} + C$
== 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:
== Also see ==
* Primitive of $\dfrac {x^3} {\sqrt {x^2 + a^2} }$
* Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.240$
Category:Primitives involving Root of a squared minus x squared
Primitive of x cubed over Root of a squared minus x squared cubed

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {a^2 - x^2} }^3} = \sqrt {a^2 - x^2} + \frac {a^2} {\sqrt {a^2 - x^2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^3} {\paren {\sqrt {x^2 + a^2} }^3}$
* Primitive of $\dfrac {x^3} {\paren {\sqrt {x^2 - a^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.254$
Category:Primitives involving Root of a squared minus x squared
Primitive of x cubed over Root of x squared minus a squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 - a^2} } = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + a^2 \sqrt {x^2 - a^2} + C$
for $\size x > a$.
== 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:
== Also see ==
* Primitive of $\dfrac {x^3} {\sqrt {x^2 + a^2} }$
* Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.212$
Category:Primitives involving Root of x squared minus a squared
Primitive of x cubed over Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 - a^2} }^3} = \sqrt {x^2 - a^2} - \frac {a^2} {\sqrt {x^2 - a^2} } + C$
for $\size x > a$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^3} {\paren {\sqrt {x^2 + a^2} }^3}$
* Primitive of $\dfrac {x^3} {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.226$
Category:Primitives involving Root of x squared minus a squared
Primitive of x cubed over Root of x squared plus a squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\sqrt {x^2 + a^2} } = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 - a^2 \sqrt {x^2 + a^2} + C$
== 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:
== Also see ==
* Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$
* Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.185$
Category:Primitives involving Root of x squared plus a squared
Primitive of x cubed over Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {\sqrt {x^2 + a^2} }^3} = \sqrt {x^2 + a^2} + \frac {a^2} {\sqrt {x^2 + a^2} } + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^3} {\paren {\sqrt {x^2 - a^2} }^3}$
* Primitive of $\dfrac {x^3} {\paren {\sqrt {a^2 - x^2} }^3}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.199$
Category:Primitives involving Root of x squared plus a squared
Primitive of x cubed over a squared minus x squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {a^2 - x^2} = -\frac {x^2} 2 - \frac {a^2} 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {x^3} {x^2 - a^2}$
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.166$
Category:Primitives involving a squared minus x squared
Primitive of x cubed over a squared minus x squared squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {a^2 - x^2}^2} = \frac {a^2} {2 \paren {a^2 - x^2} } + \frac 1 2 \map \ln {a^2 - x^2} + C$
for $x^2 < a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^3} {\paren {x^2 - a^2}^2}$
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.173$
Category:Primitives involving a squared minus x squared
Primitive of x cubed over a x + b

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {a x + b} = \frac {\paren {a x + b}^3} {3 a^4} - \frac {3 b \paren {a x + b}^2} {2 a^4} - \frac {3 b^2 \paren {a x + b} } {a^4} + \frac {b^3} {a^4} \ln \size {a x + b} + C$
== Proof ==
Put $u = a x + b$.
Then:
Then:
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.62$
Category:Primitives involving a x + b‎
Primitive of x cubed over a x + b cubed

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^3} = \frac x {a^3} - \frac {3 b^2} {a^4 \paren {a x + b} } + \frac {b^3} {2 a^4 \paren {a x + b}^2} - \frac {3 b} {a^4} \ln \size {a x + b} + C$
== Proof ==
Put $u = a x + b$.
Then:
Then:
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.76$
Category:Primitives involving a x + b‎
Primitive of x cubed over a x + b squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {a x + b}^2} = \frac {\paren {a x + b}^2} {2 a^4} - \frac {3 b \paren {a x + b} } {a^4} + \frac {b^3} {a^4 \paren {a x + b} } + \frac {3 b^2} {a^4} \ln \size {a x + b} + C$
== Proof ==
Put $u = a x + b$.
Then:
Then:
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.69$
Category:Primitives involving a x + b‎
Primitive of x cubed over x fourth minus a fourth

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {x^4 - a^4} = \frac {\ln \size {x^4 - a^4} } 4 + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^4 \pm a^4$: $14.321$
Category:Primitives involving x to the fourth plus or minus a to the fourth
Primitive of x cubed over x fourth plus a fourth

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {x^4 + a^4} = \frac {\map \ln {x^4 + a^4} } 4 + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $x^4 \pm a^4$: $14.314$
Category:Primitives involving x to the fourth plus or minus a to the fourth
Category:Primitive of x cubed over x fourth plus a fourth
Primitive of x cubed over x fourth plus a fourth/Proof 2

== Theorem ==
== Proof ==
From Primitive of $\dfrac {x^{n - 1} } {x^n + a^n}$:
:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$
So:
Category:Primitive of x cubed over x fourth plus a fourth
Primitive of x cubed over x squared minus a squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {x^2 - a^2} = \frac {x^2} 2 + \frac {a^2} 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac {x^3} {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.147$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.4.$
Category:Primitives involving x squared minus a squared
Primitive of x cubed over x squared minus a squared squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {x^2 - a^2}^2} = \frac {-a^2} {2 \paren {x^2 - a^2} } + \frac 1 2 \map \ln {x^2 - a^2} + C$
for $x^2 > a^2$.
== Proof ==
== Also see ==
* Primitive of $\dfrac {x^3} {\paren {a^2 - x^2}^2}$
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.154$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.11.$
Category:Primitives involving x squared minus a squared
Primitive of x cubed over x squared plus a squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {x^2 + a^2} = \frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 + a^2} + C$
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.128$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.4.$
Category:Primitives involving x squared plus a squared
Primitive of x cubed over x squared plus a squared squared

== Theorem ==
:$\ds \int \frac {x^3 \rd x} {\paren {x^2 + a^2}^2} = \frac {a^2} {2 \paren {x^2 + a^2} } + \frac 1 2 \map \ln {x^2 + a^2} + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.135$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.11.$
Category:Primitive of x cubed over x squared plus a squared squared
Category:Primitives involving x squared plus a squared
Primitive of x fourth by Cosine of a x

== Theorem ==
:$\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$
where $C$ is an arbitrary constant.
== Proof ==
== Also see ==
* Primitive of $x^2 \cos a x$
* Primitive of $x^2 \sin a x$
* Primitive of $x^6 \cos a x$
Category:Primitives involving Cosine Function
Primitive of x over 1 minus Cosine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {1 - \cos a x} = \frac {-x} a \cot \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sin \frac {a x} 2} + C$
== 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 $\dfrac x {1 - \sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.385$
Category:Primitives involving Cosine Function
Primitive of x over 1 minus Sine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$
== 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:
Then:
Putting it all together:
:$\ds \int \frac {x \rd x} {1 - \sin a x} = \frac x a \map \tan {\frac \pi 4 + \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 - \frac {a x} 2} } + C$
== Also see ==
* Primitive of $\dfrac x {1 - \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.355$
Category:Primitives involving Sine Function
Primitive of x over 1 plus Cosine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {1 + \cos a x} = \frac x a \tan \frac {a x} 2 + \frac 2 {a^2} \ln \size {\cos \frac {a x} 2} + C$
== 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 $\dfrac x {1 + \sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.387$
Category:Primitives involving Cosine Function
Primitive of x over 1 plus Sine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {1 + \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$
== 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:
Then:
Putting it all together:
:$\ds \int \frac {x \rd x} {1 - \sin a x} = -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C$
== Also see ==
* Primitive of $\dfrac x {1 + \cos a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.357$
Category:Primitives involving Sine Function
Primitive of x over Cosine of a x

== Theorem ==
where $E_n$ denotes the $n$th Euler number.
== Proof ==
== Also see ==
* Primitive of $\dfrac x {\sin a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cos a x$: $14.376$
Category:Primitives involving Cosine Function
Primitive of x over Cube of Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int \frac {x \rd x} {\paren {\sqrt {a x^2 + b x + c} }^3} = \frac {2 \paren {b x + 2 c} } {\paren {b^2 - 4 a c} \sqrt {a x^2 + b x + c} } + C$ 
== Proof ==
First:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.291$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x over Hyperbolic Cosine of a x

== Theorem ==
where $E_{2 n}$ denotes the $2 n$th Euler number.
== Proof ==
== Also see ==
* Primitive of $\dfrac x {\sinh a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.568$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x over Hyperbolic Cosine of a x minus 1

== Theorem ==
:$\ds \int \frac {x \rd x} {\cosh a x - 1} = -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C$
== 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 $\dfrac x {\cosh a x + 1}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.578$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x over Hyperbolic Cosine of a x plus 1

== Theorem ==
:$\ds \int \frac {x \rd x} {\cosh a x + 1} = \frac x a \tanh \frac {a x} 2 - \frac 2 {a^2} \ln \size {\cosh \frac {a x} 2} + C$
== 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 $\dfrac x {\cosh a x - 1}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.577$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x over Hyperbolic Sine of a x

== Theorem ==
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac x {\cosh a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.546$
Category:Primitives involving Hyperbolic Sine Function
Primitive of x over Power of Cosine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {\cos^n a x} = \frac {x \sin a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x} + C$
== 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:
This leads to:
== Also see ==
* Primitive of $\dfrac x {\sin^n a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.398$
* : $\S 17$: Tables of Special Indefinite Integrals: $(18)$ Integrals Involving $\cos a x$: $17.18.30.$
Category:Primitives involving Cosine Function
Primitive of x over Power of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {\cosh^n a x} = \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} a x} + C$
== Proof ==
== Also see ==
* Primitive of $\dfrac x {\sinh^n a x}$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.589$
* : $\S 17$: Tables of Special Indefinite Integrals: $(28)$ Integrals Involving $\cosh a x$: $17.28.16.$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x over Power of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x}$
== Proof ==
== Also see ==
* Primitive of $\dfrac x {\cosh^n a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.561$
* : $\S 17$: Tables of Special Indefinite Integrals: $(27)$ Integrals Involving $\sinh a x$: $17.27.16.$
Category:Primitives involving Hyperbolic Sine Function
Primitive of x over Power of Sine of a x

== Theorem ==
:$\ds \int \frac {x \rd x} {\sin^n a x} = \frac {-x \cos a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sin^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sin^{n - 2} a x}$
== 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:
This leads to:
== Also see ==
* Primitive of $\dfrac x {\cos^n a x}$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.368$
* : $\S 17$: Tables of Special Indefinite Integrals: $(17)$ Integrals Involving $\sin a x$: $17.17.30.$
Category:Primitives involving Sine Function
Primitive of x over Power of a squared minus x squared

== Theorem ==
:$\ds \int \frac {x \rd x} {\paren {a^2 - x^2}^n} = \frac 1 {2 \paren {n - 1} \paren {a^2 - x^2}^{n - 1} }$
for $x^2 < a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac x {\paren {x^2 - a^2}^n}$
== Sources ==
* : $\S 14$: Integrals involving $a^2 - x^2$, $x^2 < a^2$: $14.178$
Category:Primitives involving a squared minus x squared
Primitive of x over Power of x squared minus a squared

== Theorem ==
:$\ds \int \frac {x \rd x} {\paren {x^2 - a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 - a^2}^{n - 1} }$
for $x^2 > a^2$.
== Proof ==
Let:
== Also see ==
* Primitive of $\dfrac x {\paren {a^2 - x^2}^n}$
== Sources ==
* : $\S 14$: Integrals involving $x^2 - a^2$, $x^2 > a^2$: $14.159$
* : $\S 17$: Tables of Special Indefinite Integrals: $(7)$ Integrals Involving $x^2 - a^2$, $x^2 > a^2$: $17.7.16.$
Category:Primitives involving x squared minus a squared
Primitive of x over Power of x squared plus a squared

== Theorem ==
:$\ds \int \frac {x \rd x} {\paren {x^2 + a^2}^n} = \frac {-1} {2 \paren {n - 1} \paren {x^2 + a^2}^{n - 1} }$
== Proof ==
Let:
== Sources ==
* : $\S 14$: Integrals involving $x^2 + a^2$: $14.140$
* : $\S 17$: Tables of Special Indefinite Integrals: $(6)$ Integrals Involving $x^2 + a^2$: $17.6.16.$
Category:Primitives involving x squared plus a squared
Primitive of x over Root of 2 a x minus x squared

== Theorem ==
:$\ds \int \dfrac x {\sqrt {2 a x - x^2} } \rd x = a \arcsin \dfrac {x - a} a - \sqrt {2 a x - x^2} + C$
== 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: $54$.
Category:Primitives of Roots of Quadratic Functions