# Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form

## Theorem

$\displaystyle \int \operatorname{csch} x \ \mathrm d x = -2 \coth^{-1} \left({e^x}\right) + C$

## Proof

Let:

 $\displaystyle \int \operatorname{csch} x \ \mathrm d x$ $=$ $\displaystyle \int \frac 2 {e^x - e^{-x} } \ \mathrm d x$ Definition of Hyperbolic Cosecant $\displaystyle$ $=$ $\displaystyle \int \frac {2 e^x} {e^{2 x} - 1} \ \mathrm d x$ multiplying top and bottom by $e^x$

Let:

 $\displaystyle u$ $=$ $\displaystyle e^x$ $\displaystyle \implies \ \$ $\displaystyle u'$ $=$ $\displaystyle e^x$ Derivative of Exponential Function

Then:

 $\displaystyle \int \operatorname{csch} x \ \mathrm d x$ $=$ $\displaystyle \int \frac {2 \ \mathrm d u} {u^2 - 1}$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle -2 \coth^{-1} u + C$ Primitive of Reciprocal of $x^2 - a^2$: $\coth^{-1}$ form $\displaystyle$ $=$ $\displaystyle -2 \coth^{-1} \left({e^x}\right) + C$ Definition of $u$

$\blacksquare$

## Sources

(in which a mistake apppears)