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

== Theorem ==
:$\ds \int \sinh a x \rd x = \frac {\cosh a x} a + C$
== Proof ==
== Also see ==
* Primitive of $\cosh a x$
* Primitive of $\tanh a x$
* Primitive of $\coth a x$
* Primitive of $\sech a x$
* Primitive of $\csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.540$
* : Back endpapers: A Brief Table of Integrals: $113$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(27)$ Integrals Involving $\sinh a x$: $17.27.1.$
Category:Primitives involving Hyperbolic Sine Function
Primitive of Hyperbolic Sine of a x by Cosine of p x

== Theorem ==
:$\ds \int \sinh a x \cos p x \rd x = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x \sin p x$
* Primitive of $\cosh a x \sin p x$
* Primitive of $\cosh a x \cos p x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.552$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Cosine Function
Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

== Theorem ==
:$\ds \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
== Also see ==
* Primitive of $\sinh a x \sinh p x$
* Primitive of $\cosh a x \cosh p x$
* Primitive of $\sinh p x \cosh q x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.590$
* : $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.1.$
Category:Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 4

== 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 Hyperbolic Sine of a x by Hyperbolic Cosine of a x
Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x

== Theorem ==
:$\ds \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$
== Proof ==
== Also see ==
* Primitive of $\cosh a x \cosh p x$
* Primitive of $\sinh a x \cosh a x$
* Primitive of $\sinh p x \cosh q x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.550$
Category:Primitives involving Hyperbolic Sine Function
Primitive of Hyperbolic Sine of a x by Sine of p x

== Theorem ==
:$\ds \int \sinh a x \sin p x \rd x = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x \cos p x$
* Primitive of $\cosh a x \sin p x$
* Primitive of $\cosh a x \cos p x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.551$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Sine Function
Primitive of Hyperbolic Sine of a x over Power of x

== Theorem ==
:$\ds \int \frac {\sinh a x \rd x} {x^n} = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }$
== 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 $\dfrac {\cosh a x} {x^n}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.559$
Category:Primitives involving Hyperbolic Sine Function
Primitive of Hyperbolic Sine of a x over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\tanh a x} x$
* Primitive of $\dfrac {\coth a x} x$
* Primitive of $\dfrac {\sech a x} x$
* Primitive of $\dfrac {\csch a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.543$
Category:Primitives involving Hyperbolic Sine Function
Primitive of Hyperbolic Sine of a x over x squared

== Theorem ==
:$\ds \int \frac {\sinh a x \ \d x} {x^2} = -\frac {\sinh a x} x + a \int \frac {\cosh a x \rd x} x$
== 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:
== Also see ==
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\cosh a x} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x$: $14.544$
Category:Primitives involving Hyperbolic Sine Function
Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x

== Theorem ==
:$\ds \int \sinh p x \cosh q x \rd x = \frac {\map \cosh {p + q} x} {2 \paren {p + q} } + \frac {\map \cosh {p - q} x} {2 \paren {p - q} } + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x \sinh p x$
* Primitive of $\sinh a x \cosh a x$
* Primitive of $\cosh a x \cosh p x$
== Sources ==
* : $\S 14$: Integrals involving $\sinh a x $ and $\cosh a x$: $14.591$
* : $\S 17$: Tables of Special Indefinite Integrals: $(29)$ Integrals Involving $\sinh a x $ and $\cosh a x$: $17.29.2.$
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Primitive of Hyperbolic Tangent Function

== Theorem ==
:$\ds \int \tanh x \rd x = \map \ln {\cosh x} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of Hyperbolic Sine Function
* Primitive of Hyperbolic Cosine Function
* Primitive of Hyperbolic Cotangent Function
* Primitive of Hyperbolic Secant Function
* Primitive of Hyperbolic Cosecant Function
== Sources ==
* : Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable
* : Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxv)}$
* : $\S 14$: General Rules of Integration: $14.27$
* : Chapter $5$. Integrals: Integrals of Special Functions: $17$
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix $2$: Table of derivatives and integrals of common functions: Hyperbolic functions
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.27.$
Category:Primitives of Hyperbolic Functions
Category:Hyperbolic Tangent Function
Category:Primitive of Hyperbolic Tangent Function
Primitive of Hyperbolic Tangent of a x

== Theorem ==
:$\ds \int \tanh a x \rd x = \frac {\map \ln {\cosh a x} } a + C$
== Proof ==
== Also see ==
* Primitive of $\sinh a x$
* Primitive of $\cosh a x$
* Primitive of $\coth a x$
* Primitive of $\sech a x$
* Primitive of $\csch a x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.604$
* : Back endpapers: A Brief Table of Integrals: $123$.
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.1.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Hyperbolic Tangent of a x over x

== Theorem ==
where $B_k$ denotes the $k$th Bernoulli number.
== Proof ==
== Also see ==
* Primitive of $\dfrac {\sinh a x} x$
* Primitive of $\dfrac {\cosh a x} x$
* Primitive of $\dfrac {\coth a x} x$
* Primitive of $\dfrac {\sech a x} x$
* Primitive of $\dfrac {\csch a x} x$
== Sources ==
* : $\S 14$: Integrals involving $\tanh a x$: $14.612$
* : $\S 17$: Tables of Special Indefinite Integrals: $(30)$ Integrals Involving $\tanh a x$: $17.30.6.$
Category:Primitives involving Hyperbolic Tangent Function
Primitive of Inverse Hyperbolic Cosecant of x over a

== Theorem ==
:$\ds \int \arcsch \frac x a \rd x = \begin {cases}
x \arcsch \dfrac x a + a \arsinh \dfrac x a + C & : x > 0 \\ \\
x \arcsch \dfrac x a - a \arsinh \dfrac x a + C & : x < 0
\end {cases}$
== 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 $\arsinh \dfrac x a$
* Primitive of $\arcosh \dfrac x a$
* Primitive of $\artanh \dfrac x a$
* Primitive of $\arcoth \dfrac x a$
* Primitive of $\arsech \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.669$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.14.$
Category:Primitives involving Inverse Hyperbolic Cosecant Function
Primitive of Inverse Hyperbolic Cosecant of x over a over x

== Theorem ==
== Proof ==
For $\size x < a$:
For $0 < x < a$:
For $-a < x < 0$:
== Also see ==
* Primitive of $\dfrac 1 x \arsinh \dfrac x a$
* Primitive of $\dfrac 1 x \arcosh \dfrac x a$
* Primitive of $\dfrac 1 x \artanh \dfrac x a$
* Primitive of $\dfrac 1 x \arcoth \dfrac x a$
* Primitive of $\dfrac 1 x \arsech \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.671$
Category:Primitives involving Inverse Hyperbolic Cosecant Function
Primitive of Inverse Hyperbolic Cosine Function

== Theorem ==
:$\ds \int \arcosh x \rd x = x \arcosh x - \sqrt {x^2 - 1} + C$
== Proof ==
== Also see ==
* Primitive of $\arsinh x$
* Primitive of $\artanh x$
* Primitive of $\arcoth x$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitives involving Inverse Hyperbolic Cosine Function
Primitive of Inverse Hyperbolic Cosine Function/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 Inverse Hyperbolic Cosine Function
Primitive of Inverse Hyperbolic Cosine of x over a

== Theorem ==
:$\ds \int \arcosh \frac x a \rd x = x \arcosh \dfrac x a - \sqrt {x^2 - a^2} + C$
where $\arcosh$ denotes the real area hyperbolic cosine.
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\arsinh \dfrac x a$
* Primitive of $\artanh \dfrac x a$
* Primitive of $\arcoth \dfrac x a$
* Primitive of $\arsech \dfrac x a$
* Primitive of $\arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.651$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.4.$
Category:Primitive of Inverse Hyperbolic Cosine of x over a
Category:Primitive of Inverse Hyperbolic Cosine Function
Category:Primitives involving Inverse Hyperbolic Cosine Function
Primitive of Inverse Hyperbolic Cosine of x over a/Corollary

== Theorem ==
:$\ds \int -\cosh^{-1} \frac x a \rd x = x \paren {-\cosh^{-1} \dfrac x a} + \sqrt {x^2 - a^2} + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.651$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.4.$
Category:Primitive of Inverse Hyperbolic Cosine of x over a
Primitive of Inverse Hyperbolic Cosine of x over a/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 Inverse Hyperbolic Cosine of x over a
Primitive of Inverse Hyperbolic Cosine of x over a over x

== Theorem ==
where $\arcosh$ denotes the real area hyperbolic cosine.
=== Corollary ===
== Proof ==
For $\arcosh \dfrac x a > 0$:
== Also see ==
* Primitive of $\dfrac 1 x \arsinh \dfrac x a$
* Primitive of $\dfrac 1 x \artanh \dfrac x a$
* Primitive of $\dfrac 1 x \arcoth \dfrac x a$
* Primitive of $\dfrac 1 x \arsech \dfrac x a$
* Primitive of $\dfrac 1 x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.654$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.6.$
Category:Primitives involving Inverse Hyperbolic Cosine Function
Category:Primitive of Inverse Hyperbolic Cosine of x over a over x
Primitive of Inverse Hyperbolic Cosine of x over a over x/Corollary

== Theorem ==
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.654$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.6.$
Category:Primitive of Inverse Hyperbolic Cosine of x over a over x
Primitive of Inverse Hyperbolic Cosine of x over a over x squared

== Theorem ==
:$\ds \int \frac 1 {x^2} \arcosh \dfrac x a \rd x = -\frac 1 x \arcosh \dfrac x a + \frac 1 a \arcsec \size {\frac x a} + C$
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 $\dfrac {\arsinh \frac x a} {x^2}$
* Primitive of $\dfrac {\artanh \frac x a} {x^2}$
* Primitive of $\dfrac {\arcoth \frac x a} {x^2}$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.655$
Category:Primitives involving Inverse Hyperbolic Cosine Function
Category:Primitive of Inverse Hyperbolic Cosine of x over a over x squared
Primitive of Inverse Hyperbolic Cosine of x over a over x squared/Corollary

== Theorem ==
:$\ds \int \frac 1 {x^2} \paren {-\cosh^{-1} \dfrac x a} \rd x = -\frac 1 x \paren {-\cosh^{-1} \dfrac x a} - \frac 1 a \arcsec \size {\frac x a} + C$
where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.655$
Category:Primitive of Inverse Hyperbolic Cosine of x over a over x squared
Primitive of Inverse Hyperbolic Cosine of x over a over x squared/Mistake

== Source Work ==
:Chapter $14$: Indefinite Integrals
::Integrals involving Inverse Hyperbolic Functions: $14.655$
This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).
== Mistake ==
:$\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = \dfrac {-\map {\cosh^{-1} } {x / a} } x \mp \dfrac 1 a \map \ln {\dfrac {a + \sqrt {x^2 + a^2} } x}$ $\sqbrk {- \text { if } \map {\cosh^{-1} } {x / a} > 0, + \text { if } \map {\cosh^{-1} } {x / a} < 0}$
== Correction ==
As demonstrated in Primitive of $\dfrac {\map {\cosh^{-1} } {x / a} } {x^2}$, this is incorrect.
It should be:
:$\ds \int \frac {\map {\cosh^{-1} } {x / a} } {x^2} \rd x = -\frac 1 x \cosh^{-1} \dfrac x a \pm \frac 1 a \arcsec \size {\frac x a}$ $\sqbrk {+ \text { if } \map {\cosh^{-1} } {x / a} > 0, - \text { if } \map {\cosh^{-1} } {x / a} < 0}$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.655$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Primitive of Inverse Hyperbolic Cotangent Function

== Theorem ==
:$\ds \int \arcoth x \rd x = x \arcoth x + \frac {\map \ln {x^2 - 1} } 2 + C$
for $x^2 > 1$.
== Proof ==
== Also presented as ==
== Also see ==
* Primitive of $\arsinh x$
* Primitive of $\arcosh x$
* Primitive of $\artanh x$
== Sources ==
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Inverse Hyperbolic Cotangent Function
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Cotangent Function/Also presented as

== Primitive of Inverse Hyperbolic Tangent Function: Also presented as ==
This result can also be presented as:
:$\ds \int \arcoth x \rd x = x \arcoth x + \ln \sqrt {x^2 - 1} + C$
== Proof ==
From Primitive of $\artanh x$:
:$\ds \int \arcoth x \rd x = x \arcoth x + \frac {\map \ln {x^2 - 1} } 2 + C$
The result follows by observing:
:$\dfrac 1 2 \log x = \log \sqrt x$
from Logarithm of Power.
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Cotangent Function/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 Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Cotangent of x over a

== Theorem ==
:$\ds \int \arcoth \frac x a \rd x = x \arcoth \dfrac x a + \frac {a \map \ln {x^2 - a^2} } 2 + C$
for $x^2 > a^2$.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\arsinh \dfrac x a$
* Primitive of $\arcosh \dfrac x a$
* Primitive of $\artanh \dfrac x a$
* Primitive of $\arsech \dfrac x a$
* Primitive of $\arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.661$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.10.$
Category:Primitive of Inverse Hyperbolic Cotangent of x over a
Category:Primitive of Inverse Hyperbolic Cotangent Function
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Cotangent of x over a/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 Inverse Hyperbolic Cotangent of x over a
Primitive of Inverse Hyperbolic Cotangent of x over a over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arsinh \dfrac x a$
* Primitive of $\dfrac 1 x \arcosh \dfrac x a$
* Primitive of $\dfrac 1 x \artanh \dfrac x a$
* Primitive of $\dfrac 1 x \arsech \dfrac x a$
* Primitive of $\dfrac 1 x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.664$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.12.$
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Cotangent of x over a over x squared

== Theorem ==
:$\ds \int \frac 1 {x^2} \arcoth \dfrac x a \rd x = -\frac 1 x \arcoth \dfrac x a + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C$
== 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 $\dfrac 1 {x^2} \arsinh \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \arcosh \dfrac x a$
* Primitive of $\dfrac 1 {x^2} \artanh \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.665$
Category:Primitives involving Inverse Hyperbolic Cotangent Function
Primitive of Inverse Hyperbolic Secant of x over a

== Theorem ==
:$\ds \int \arsech \frac x a \rd x = x \arsech \dfrac x a + a \arcsin \dfrac x a + C$
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 $\arsinh \dfrac x a$
* Primitive of $\arcosh \dfrac x a$
* Primitive of $\artanh \dfrac x a$
* Primitive of $\arcoth \dfrac x a$
* Primitive of $\arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.666$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.13.$
Category:Primitives involving Inverse Hyperbolic Secant Function
Category:Primitive of Inverse Hyperbolic Secant of x over a
Primitive of Inverse Hyperbolic Secant of x over a/Corollary

== Theorem ==
:$\ds \int \sech^{-1} \frac x a \rd x = -x \paren {-\sech^{-1} \dfrac x a} - a \arcsin \dfrac x a + C$
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.666$
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.13.$
Category:Primitive of Inverse Hyperbolic Secant of x over a
Primitive of Inverse Hyperbolic Secant of x over a over x

== Theorem ==
where $\arsech$ denotes the real area hyperbolic secant.
=== Corollary ===
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arsinh \dfrac x a$
* Primitive of $\dfrac 1 x \arcosh \dfrac x ax$
* Primitive of $\dfrac 1 x \artanh \dfrac x a$
* Primitive of $\dfrac 1 x \arcoth \dfrac x a$
* Primitive of $\dfrac 1 x \arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.668$
Category:Primitives involving Inverse Hyperbolic Secant Function
Category:Primitive of Inverse Hyperbolic Secant of x over a over x
Primitive of Inverse Hyperbolic Secant of x over a over x/Corollary

== Theorem ==
where $-\sech^{-1}$ denotes the negative branch of the real inverse hyperbolic secant multifunction.
== Proof ==
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.668$
Category:Primitive of Inverse Hyperbolic Secant of x over a over x
Primitive of Inverse Hyperbolic Secant of x over a over x/Lemma

== Lemma for Primitive of Inverse Hyperbolic Secant of x over a over x ==
:$\map {\ln^2} {\dfrac x {2 a} } = \map \ln {\dfrac a x} \map \ln {\dfrac {4 a} x} + \map {\ln^2} 2$
== Proof ==
$$\ln \left(\frac{a}{x}\right) \ln \left(\frac{4 a}{x}\right)-\ln ^2\left(\frac{x}{2 a}\right)=
\ln \left(\frac{a}{x}\right) \left[\ln \left(\frac{a}{x}\right)+2 \log (2)\right]-\left[\ln \left(\frac{a}{x}\right)+\ln (2)\right]^2
=-\ln ^2(2)$$
Then:
Hence the result.
Category:Primitive of Inverse Hyperbolic Secant of x over a over x
Primitive of Inverse Hyperbolic Sine Function

== Theorem ==
:$\ds \int \arsinh x \rd x = x \arsinh x - \sqrt {x^2 + 1} + C$
== Proof ==
== Also see ==
* Primitive of $\arcosh x$
* Primitive of $\artanh x$
* Primitive of $\arcoth x$
== Sources ==
* : $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
* : Appendix: Table $2$: Integrals
* : Appendix: Table $2$: Integrals
Category:Primitive of Inverse Hyperbolic Sine Function
Category:Primitives involving Inverse Hyperbolic Sine Function
Primitive of Inverse Hyperbolic Sine Function/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 Inverse Hyperbolic Sine Function
Primitive of Inverse Hyperbolic Sine of x over a

== Theorem ==
:$\ds \int \arsinh \frac x a \rd x = x \arsinh \frac x a - \sqrt {x^2 + a^2} + C$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Primitive of $\arcosh \dfrac x a$
* Primitive of $\artanh \dfrac x a$
* Primitive of $\arcoth \dfrac x a$
* Primitive of $\arsech \dfrac x a$
* Primitive of $\arcsch \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.646$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse hyperbolic functions
* : $\S 17$: Tables of Special Indefinite Integrals: $(34)$ Integrals Involving Inverse Hyperbolic Functions: $17.34.1.$
Category:Primitive of Inverse Hyperbolic Sine of x over a
Category:Primitive of Inverse Hyperbolic Sine Function
Category:Primitives involving Inverse Hyperbolic Sine Function
Primitive of Inverse Hyperbolic Sine of x over a/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 Inverse Hyperbolic Sine of x over a