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

Theorem

$\displaystyle \int \cot^n a x \csc^2 a x \ \mathrm d x = \frac {-\cot^{n + 1} a x} {\left({n + 1}\right) a} + C$

Proof

 $\displaystyle z$ $=$ $\displaystyle \cot a x$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d z} {\mathrm d x}$ $=$ $\displaystyle -a \csc^2 a x$ Derivative of Cotangent Function: Corollary $\displaystyle \implies \ \$ $\displaystyle \int \cot^n a x \csc^2 a x \ \mathrm d x$ $=$ $\displaystyle \int \frac {-1} a z^n \ \mathrm d z$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac {-1} a \frac {z^{n + 1} } {n + 1}$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {-\cot^{n + 1} a x} {\left({n + 1}\right) a} + C$ substituting for $z$ and simplifying

$\blacksquare$