Primitive of Hyperbolic Cotangent Function

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Theorem

$\ds \int \coth x \rd x = \ln \size {\sinh x} + C$

where $\sinh x \ne 0$.


Proof

\(\ds \int \coth x \rd x\) \(=\) \(\ds \int \frac {\cosh x} {\sinh x} \rd x\) Definition of Hyperbolic Cotangent
\(\ds \) \(=\) \(\ds \int \frac {\paren {\sinh x}'} {\sinh x} \rd x\) Derivative of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \ln \size {\sinh x} + C\) Primitive of Function under its Derivative

$\blacksquare$


Also see


Sources

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