Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent Form

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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 $u$

$\blacksquare$


Also see


Sources

(in which a mistake apppears)