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

== Theorem ==
:$\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$
== 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 $x \cosh a x$
* Primitive of $x \tanh a x$
* Primitive of $x \coth a x$
* Primitive of $x \sech a x$
* Primitive of $x \csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.541$
* : Back endpapers: A Brief Table of Integrals: $119$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(27)$ Integrals Involving $\sinh a x$: $17.27.2.$
Category:Primitives involving Hyperbolic Sine Function
Primitive of x by Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.
== Proof ==
 
== Also see ==
* Primitive of $x \sinh a x$
* Primitive of $x \cosh a x$
* Primitive of $x \coth a x$
* Primitive of $x \sech a x$
* Primitive of $x \csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.610$
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.4.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of x by Inverse Hyperbolic Cosecant of x over a

== Theorem ==
:$\ds \int x \arcsch \frac x a \rd x = \begin {cases}
\dfrac {x^2} 2 \arcsch \dfrac x a + \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x > 0 \\ \\
\dfrac {x^2} 2 \arcsch \dfrac x a - \dfrac {a \sqrt {x^2 + a^2} } 2 + C & : x < 0
\end {cases}$
== 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 $x \arsinh \dfrac x a$
* Primitive of $x \arcosh \dfrac x a$
* Primitive of $x \artanh \dfrac x a$
* Primitive of $x \arcoth \dfrac x a$
* Primitive of $x \arsech \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.670$
Category:Primitives involving Inverse Hyperbolic Cosecant Function
Primitive of x by Inverse Hyperbolic Cosine of x over a

== Theorem ==
:$\ds \int x \arcosh \frac x a \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \arcosh \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $\arcosh$ denotes the real area hyperbolic cosine.
=== Corollary ===
== 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 $x \arsinh \dfrac x a$
* Primitive of $x \artanh \dfrac x a$
* Primitive of $x \arcoth \dfrac x a$
* Primitive of $x \arsech \dfrac x a$
* Primitive of $x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.652$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.5.$
Category:Primitives involving Inverse Hyperbolic Cosine Function
Category:Primitive of x by Inverse Hyperbolic Cosine of x over a
Primitive of x by Inverse Hyperbolic Cosine of x over a/Corollary

== Theorem ==
:$\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4 + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.652$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.5.$
Category:Primitive of x by Inverse Hyperbolic Cosine of x over a
Primitive of x by Inverse Hyperbolic Cotangent of x over a

== Theorem ==
:$\ds \int x \arcoth \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \arcoth \frac x a + C$
== 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 $x \arsinh \dfrac x a$
* Primitive of $x \arcosh \dfrac x a$
* Primitive of $x \artanh \dfrac x a$
* Primitive of $x \arsech \dfrac x a$
* Primitive of $x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.662$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.11.$
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of x by Inverse Hyperbolic Secant of x over a

== Theorem ==
:$\ds \int x \arsech \frac x a \rd x = \dfrac {x^2} 2 \arsech \dfrac x a - \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $\arsech$ denotes the real area hyperbolic secant.
=== Corollary ===
== 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 $x \arsinh \dfrac x a$
* Primitive of $x \arcosh \dfrac x a$
* Primitive of $x \artanh \dfrac x a$
* Primitive of $x \arcoth \dfrac x a$
* Primitive of $x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.667$
Category:Primitives involving Inverse Hyperbolic Secant Function
Category:Primitive of x by Inverse Hyperbolic Secant of x over a
Primitive of x by Inverse Hyperbolic Secant of x over a/Corollary

== Theorem ==
:$\ds \int x \paren {-\sech^{-1} \frac x a} \rd x = -\dfrac {x^2} 2 \paren {-\sech^{-1} \frac x a} + \dfrac {a \sqrt {a^2 - x^2} } 2 + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.667$
Category:Primitive of x by Inverse Hyperbolic Secant of x over a
Primitive of x by Inverse Hyperbolic Sine of x over a

== Theorem ==
:$\ds \int x \arsinh \frac x a \rd x = \paren {\frac {x^2} 2 + \frac {a^2} 4} \arsinh \frac x a - \frac {x \sqrt {x^2 + a^2} } 4 + C$
== 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 $x \arcosh \dfrac x a$
* Primitive of $x \artanh \dfrac x a$
* Primitive of $x \arcoth \dfrac x a$
* Primitive of $x \arsech \dfrac x a$
* Primitive of $x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.647$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.2.$
Category:Primitives involving Inverse Hyperbolic Sine Function
Primitive of x by Inverse Hyperbolic Tangent of x over a

== Theorem ==
:$\ds \int x \artanh \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \artanh \frac x a + C$
== 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 $x \arsinh \dfrac x a$
* Primitive of $x \arcosh \dfrac x a$
* Primitive of $x \arcoth \dfrac x a$
* Primitive of $x \arsech \dfrac x a$
* Primitive of $x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.657$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.8.$
Category:Primitives involving Inverse Hyperbolic Tangent Function
Primitive of x by Logarithm of x

== Theorem ==
:$\ds \int x \ln x \rd x = \frac {x^2} 2 \paren {\ln x - \frac 1 2} + C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 14$: Integrals involving $\ln x$: $14.526$
* : $\S 17$: Tables of Special Indefinite Integrals: $(26)$ Integrals Involving $\ln x$: $17.26.2.$
Category:Primitive of x by Logarithm of x
Category:Primitives involving Logarithm Function
Primitive of x by Logarithm of x/Proof 1

== Theorem ==
== 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:
Category:Primitive of x by Logarithm of x
Primitive of x by Logarithm of x/Proof 2

== Theorem ==
== Proof ==
From Primitive of $x^m \ln x$:
:$\ds \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \paren {\ln x - \frac 1 {m + 1} } + C$
The result follows by setting $m = 1$.
Category:Primitive of x by Logarithm of x
Primitive of x by Logarithm of x squared plus a squared

== Theorem ==
:$\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$
== Proof 1 ==
== Proof 2 ==
Category:Primitives involving Logarithm Function
Category:Primitives involving x squared plus a squared
Category:Primitive of x by Logarithm of x squared plus a squared
Primitive of x by Power of Root of a x + b

== Theorem ==
:$\ds \int x \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 4} } {a^2 \paren {m + 4} } - \frac {2 b \paren {\sqrt {a x + b} }^{m + 2} } {a^2 \paren {m + 2} } + C$
== Proof ==
Let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.100$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.17.$
Category:Primitives involving Root of a x + b
Primitive of x by Power of Root of x squared minus a squared

== Theorem ==
:$\ds \int x \paren {\sqrt {x^2 - a^2} }^n \rd x = \dfrac {\paren {\sqrt {x^2 - a^2} }^{n + 2} } {n + 2} + C$
for $n \ne -2$.
== Proof ==
Let:
Thus:
== Also see ==
For $n = -2$, use Primitive of $\dfrac x {x^2 - a^2}$.
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $40$.
Category:Primitives involving Root of x squared minus a squared
Primitive of x by Power of a x + b

== Theorem ==
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 2} } {\paren {n + 2} a^2} - \frac {b \paren {a x + b}^{n + 1} } {\paren {n + 1} a^2} + C$
where $n \ne - 1$ and $n \ne - 2$.
== Proof ==
Let $u = a x + b$.
Then:
Then:
== Also presented as ==
== Also see ==
* Primitive of $x$ over $a x + b$ for the case when $n = -1$
* Primitive of $x$ over $\paren {a x + b}^2$ for the case when $n = -2$
== Sources ==
* : $\S 14$: Integrals involving $a x + b$: $14.81$
* : $\S 17$: Tables of Special Indefinite Integrals: $(1)$ Integrals Involving $a x + b$: $17.1.15.$
Category:Primitive of x by Power of a x + b
Category:Primitives involving a x + b‎
Primitive of x by Power of a x + b/Also presented as

== Primitive of $x$ by Power of $a x + b$: Also presented as ==
This result is also seen presented in the form:
:$\ds \int x \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {a^2} \paren {\frac {a x + b} {n + 2} - \frac b {n + 1} } + C$
where $n \ne - 1$ and $n \ne - 2$.
== Also see ==
* Primitive of $x$ over $a x + b$ for the case when $n = -1$
* Primitive of $x$ over $\paren {a x + b}^2$ for the case when $n = -2$
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $7$.
Category:Primitive of x by Power of a x + b
Primitive of x by Root of 2 a x minus x squared

== Theorem ==
:$\ds \int x \sqrt {2 a x - x^2} \rd x = \frac {\paren {x + a} \paren {2 x - 3 a} \sqrt {2 a x - x^2} } 6 + \frac {a^3} 2 \arcsin \dfrac {x - a} a + C$
== Proof ==
Let $u := x - a$.
Then:
:$\dfrac {\d u} {\d x} = 1$
and:
:$x = u + a$
Then:
and we have:
The result follows after algebra.
== Sources ==
* : Front endpapers: A Brief Table of Integrals: $51$.
Category:Primitives of Roots of Quadratic Functions
Primitive of x by Root of a squared minus x squared

== Theorem ==
:$\ds \int x \sqrt {a^2 - x^2} \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^3} 3 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x \sqrt {x^2 + a^2}$
* Primitive of $x \sqrt {x^2 - a^2}$
== Sources ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $8$.
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.245$
Category:Primitives involving Root of a squared minus x squared
Primitive of x by Root of a squared minus x squared cubed

== Theorem ==
:$\ds \int x \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-\paren {\sqrt {a^2 - x^2} }^5} 5 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x \paren {\sqrt {x^2 - a^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a^2 - x^2}$: $14.259$
Category:Primitives involving Root of a squared minus x squared
Primitive of x by Root of a x + b

== Theorem ==
:$\ds \int x \sqrt {a x + b} \rd x = \frac {2 \paren {3 a x - 2 b} } {15 a^2} \sqrt {\paren {a x + b}^3}$
== Proof ==
Let:
Then:
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.90$
* : $\S 17$: Tables of Special Indefinite Integrals: $(2)$ Integrals Involving $\sqrt {a x + b}$: $17.2.7.$
Category:Primitives involving Root of a x + b
Primitive of x by Root of a x squared plus b x plus c

== Theorem ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac {b \paren {2 a x + b} \sqrt {a x^2 + b x + c} } {8 a^2} - \frac {b \paren {4 a c - b^2} } {16 a^2} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 {\sqrt {a x^2 + b x + c} }$
* For $a = 0$, see Primitive of $x \sqrt {a x + b}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.286$
Category:Primitives involving Root of a x squared plus b x plus c
Primitive of x by Root of a x squared plus b x plus c/Lemma

== Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$ ==
Let $a \in \R_{\ne 0}$.
Then:
:$\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x$
== Proof ==
Let:
So:
Category:Primitives involving Root of a x squared plus b x plus c/Lemmata
Primitive of x by Root of x squared minus a squared

== Theorem ==
:$\ds \int x \sqrt {x^2 - a^2} \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^3} 3 + C$
for $\size x \ge a$.
== Proof ==
Let:
== Also see ==
* Primitive of $x \sqrt {x^2 + a^2}$
* Primitive of $x \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.217$
Category:Primitives involving Root of x squared minus a squared
Primitive of x by Root of x squared minus a squared cubed

== Theorem ==
:$\ds \int x \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 - a^2} }^5} 5 + C$
for $\size x \ge a$.
== Proof ==
Let:
== Also see ==
* Primitive of $x \paren {\sqrt {x^2 + a^2} }^3$
* Primitive of $x \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 - a^2}$: $14.231$
Category:Primitives involving Root of x squared minus a squared
Primitive of x by Root of x squared plus a squared

== Theorem ==
:$\ds \int x \sqrt {x^2 + a^2} \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^3} 3 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x \sqrt {x^2 - a^2}$
* Primitive of $x \sqrt {a^2 - x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.190$
Category:Primitives involving Root of x squared plus a squared
Primitive of x by Root of x squared plus a squared cubed

== Theorem ==
:$\ds \int x \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^5} 5 + C$
== Proof ==
Let:
== Also see ==
* Primitive of $x \paren {\sqrt {x^2 - a^2} }^3$
* Primitive of $x \paren {\sqrt {a^2 - z^2} }^3$
== Sources ==
* : $\S 14$: Integrals involving $\sqrt {x^2 + a^2}$: $14.204$
Category:Primitives involving Root of x squared plus a squared
Primitive of x by Secant of a x

== Theorem ==
where $E_{2 n}$ is the $2 n$th Euler number.
== Proof ==
 
== Also see ==
* Primitive of $x \sin a x$
* Primitive of $x \cos a x$
* Primitive of $x \tan a x$
* Primitive of $x \cot a x$
* Primitive of $x \csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.456$
Category:Primitives involving Secant Function
Primitive of x by Sine of a x

== Theorem ==
:$\ds \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$
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 $x \cos a x$
* Primitive of $x \tan a x$
* Primitive of $x \cot a x$
* Primitive of $x \sec a x$
* Primitive of $x \csc a x$
== Sources ==
* : $\S 14$: Integrals involving $\sin a x$: $14.340$
* : Back endpapers: A Brief Table of Integrals: $78$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(17)$ Integrals Involving $\sin a x$: $17.17.2.$
Category:Primitives involving Sine Function
Primitive of x by Square of Cosecant of a x

== Theorem ==
:$\ds \int x \csc^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} + C$
== 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:
== Also see ==
* Primitive of $x \sin^2 a x$
* Primitive of $x \cos^2 a x$
* Primitive of $x \tan^2 a x$
* Primitive of $x \cot^2 a x$
* Primitive of $x \sec^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csc a x$: $14.468$
Category:Primitives involving Cosecant Function
Primitive of x by Square of Cosine of a x

== Theorem ==
:$\ds \int x \cos^2 a x \rd x = \frac {x^2} 4 + \frac {x \sin 2 a x} {4 a} + \frac {\cos 2 a x} {8 a^2} + C$
== 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 $x \sin^2 a x$
* 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 $\cos a x$: $14.378$
Category:Primitives involving Cosine Function
Primitive of x by Square of Cotangent of a x

== Theorem ==
:$\ds \int x \cot^2 a x \rd x = \frac {-x \cot a x} a + \frac 1 {a^2} \ln \size {\sin a x} - \frac {x^2} 2 + C$
== 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 $x \tan^2 a x$
* Primitive of $x \sec^2 a x$
* Primitive of $x \csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cot a x$: $14.448$
Category:Primitives involving Cotangent Function
Primitive of x by Square of Hyperbolic Cosecant of a x

== Theorem ==
:$\ds \int x \csch^2 a x \rd x = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$
== 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:
== Also see ==
* Primitive of $x \sinh^2 a x$
* Primitive of $x \cosh^2 a x$
* Primitive of $x \tanh^2 a x$
* Primitive of $x \coth^2 a x$
* Primitive of $x \sech^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\csch a x$: $14.642$
* : $\S 17$: Tables of Special Indefinite Integrals: $(33)$ Integrals Involving $\csch a x$: $17.33.5.$
Category:Primitives involving Hyperbolic Cosecant Function
Primitive of x by Square of Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int x \cosh^2 a x \rd x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$
== 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 $x \sinh^2 a x$
* Primitive of $x \tanh^2 a x$
* Primitive of $x \coth^2 a x$
* Primitive of $x \sech^2 a x$
* Primitive of $x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\cosh a x$: $14.570$
Category:Primitives involving Hyperbolic Cosine Function
Primitive of x by Square of Hyperbolic Cotangent of a x

== Theorem ==
:$\ds \int x \coth^2 a x \rd x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$
== 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:
== Also see ==
* Primitive of $x \sinh^2 a x$
* Primitive of $x \cosh^2 a x$
* Primitive of $x \tanh^2 a x$
* Primitive of $x \sech^2 a x$
* Primitive of $x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\coth a x$: $14.622$
* : $\S 17$: Tables of Special Indefinite Integrals: $(31)$ Integrals Involving $\coth a x$: $17.31.5.$
Category:Primitives involving Hyperbolic Cotangent Function
Primitive of x by Square of Hyperbolic Secant of a x

== Theorem ==
:$\ds \int x \sech^2 a x \rd x = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \size {\cosh a x} + C$
== 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:
== Also see ==
* Primitive of $x \sinh^2 a x$
* Primitive of $x \cosh^2 a x$
* Primitive of $x \tanh^2 a x$
* Primitive of $x \coth^2 a x$
* Primitive of $x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sech a x$: $14.632$
* : $\S 17$: Tables of Special Indefinite Integrals: $(32)$ Integrals Involving $\sech a x$: $17.32.5.$
Category:Primitives involving Hyperbolic Secant Function
Primitive of x by Square of Hyperbolic Sine of a x

== Theorem ==
:$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$
== 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 $x \cosh^2 a x$
* Primitive of $x \tanh^2 a x$
* Primitive of $x \coth^2 a x$
* Primitive of $x \sech^2 a x$
* Primitive of $x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.548$
Category:Primitives involving Hyperbolic Sine Function
Primitive of x by Square of Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int x \tanh^2 a x \rd x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \size {\cosh a x} + C$
== 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:
== Also see ==
* Primitive of $x \sinh^2 a x$
* Primitive of $x \cosh^2 a x$
* Primitive of $x \coth^2 a x$
* Primitive of $x \sech^2 a x$
* Primitive of $x \csch^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.611$
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.5.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of x by Square of Logarithm of x

== Theorem ==
:$\ds \int x \ln^2 x \rd x = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C$
== 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 ==
* : $\text {II}$. Calculus: Exercises $\text {XIV}$: $25$.
Category:Primitives involving Logarithm Function
Primitive of x by Square of Secant of a x

== Theorem ==
:$\ds \int x \sec^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} + C$
== 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:
== Also see ==
* Primitive of $x \sin^2 a x$
* Primitive of $x \cos^2 a x$
* Primitive of $x \tan^2 a x$
* Primitive of $x \cot^2 a x$
* Primitive of $x \csc^2 a x$
== Sources ==
* : $\S 14$: Integrals involving $\sec a x$: $14.458$
Category:Primitives involving Secant Function