# Primitive of Cube of Cotangent of a x

## Theorem

$\displaystyle \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$

## Proof

 $\displaystyle \int \cot^3 x \rd x$ $=$ $\displaystyle \int \cot a x \cot^2 a x \rd x$ $\displaystyle$ $=$ $\displaystyle \int \cot a x \paren {\csc^2 a x - 1} \rd x$ Difference of Squares of Cosecant and Cotangent $\displaystyle$ $=$ $\displaystyle \int \cot a x \csc^2 a x \rd x - \int \cot a x \rd x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac {-\cot^2 a x} {2 a} - \int \cot a x \rd x + C$ Primitive of $\cot^n a x \csc^2 a x$: $n = 1$ $\displaystyle$ $=$ $\displaystyle \frac {-\cot^2 a x} {2 a} - \paren {\frac {\ln \size {\sin a x} } a} + C$ Primitive of $\cot a x$ $\displaystyle$ $=$ $\displaystyle \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$ simplifying

$\blacksquare$