Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$


Proof

\(\ds z\) \(=\) \(\ds \csch a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds -a \csch a x \coth a x\) Derivative of $\csch a x$
\(\ds \leadsto \ \ \) \(\ds \int \csch^n a x \coth a x \rd x\) \(=\) \(\ds \int \frac {-z^{n - 1} \rd z} a\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac {-1} a \int z^{n - 1} \rd z\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac {-1} a \frac {z^n} n\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-\csch^n a x} {n a} + C\) substituting for $z$

$\blacksquare$


Also see


Sources