# Primitive of Power of Cotangent of a x by Square of Cosecant of a x

## Theorem

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

## Proof

 $\ds z$ $=$ $\ds \cot a x$ $\ds \leadsto \ \$ $\ds \frac {\d z} {\d x}$ $=$ $\ds -a \csc^2 a x$ Derivative of Cotangent Function: Corollary $\ds \leadsto \ \$ $\ds \int \cot^n a x \csc^2 a x \rd x$ $=$ $\ds \int \frac {-1} a z^n \rd z$ Integration by Substitution $\ds$ $=$ $\ds \frac {-1} a \frac {z^{n + 1} } {n + 1}$ Primitive of Power $\ds$ $=$ $\ds \frac {-\cot^{n + 1} a x} {\paren {n + 1} a} + C$ substituting for $z$ and simplifying

$\blacksquare$