Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x

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Theorem

$\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$


Proof

\(\ds \frac \d {\d x} \coth a x\) \(=\) \(\ds -a \csch^2 a x\) Derivative of $\coth a x$
\(\ds \leadsto \ \ \) \(\ds \int \frac {-a \csch^2 a x \rd x} {\coth a x}\) \(=\) \(\ds \ln \size {\coth a x} + C\) Primitive of Function under its Derivative
\(\ds \leadsto \ \ \) \(\ds \int \frac {\csch^2 a x \rd x} {\coth a x}\) \(=\) \(\ds \frac {-1} a \ln \size {\coth a x} + C\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


Sources