Primitive of Power of Cotangent of a x

Theorem

$\ds \int \cot^n a x \rd x = \frac {-\cot^{n - 1} a x} {\paren {n - 1} a} - \int \cot^{n - 2} a x \rd x$

for $n \ne 1$.

$\ds \int \cot^n a x \rd x$

$=$

$\ds \int \cot^{n - 2} a x \cot^2 a x \rd x$

$\ds $

$=$

$\ds \int \cot^{n - 2} a x \paren {\csc^2 a x - 1} \rd x$

$\ds $

$=$

$\ds \int \cot^{n - 2} a x \csc^2 a x \rd x - \int \cot^{n - 2} \rd x$

$\ds $

$=$

$\ds \frac {-\cot^{n - 1} a x} {\paren {n - 1} a} - \int \cot^{n - 2} \rd x$

$\blacksquare$