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

## Theorem

$\displaystyle \int \frac {\csc^2 a x \ \mathrm d x} {\cot a x} = \frac {-\ln \left\vert{\cot a x}\right\vert} a + C$

## Proof

 $\displaystyle \frac {\mathrm d} {\mathrm d x} \cot x$ $=$ $\displaystyle -\csc^2 x$ Derivative of Cotangent Function $\displaystyle \implies \ \$ $\displaystyle \int \frac {\csc^2 x \ \mathrm d x} {\cot x}$ $=$ $\displaystyle -\ln \left\vert{\cot a x}\right\vert + C$ Primitive of Function under its Derivative $\displaystyle \implies \ \$ $\displaystyle \int \frac {\csc^2 a x \ \mathrm d x} {\cot a x}$ $=$ $\displaystyle \frac 1 a \left({-\ln \left\vert{\cot a x}\right\vert}\right) + C$ Primitive of Function of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac {-\ln \left\vert{\cot a x}\right\vert} a + C$ simplifying

$\blacksquare$