# Primitive of Reciprocal of 1 minus Cosine of a x

## Theorem

$\displaystyle \int \frac {\mathrm d x} {1 - \cos a x} = \frac {-1} a \cot \frac {a x} 2 + C$

## Proof

 $\displaystyle u$ $=$ $\displaystyle \tan \frac x 2$ $\displaystyle \implies \ \$ $\displaystyle \int \frac {\mathrm d x} {1 - \cos x}$ $=$ $\displaystyle \int \frac {\dfrac {2 \ \mathrm d u} {1 + u^2} } {1 - \dfrac {1 - u^2} {1 + u^2} }$ Weierstrass Substitution $\displaystyle$ $=$ $\displaystyle \int \frac {2 \ \mathrm d u} {1 + u^2 - \left({1 - u^2}\right)}$ multiplying top and bottom by $1 + u^2$ $\displaystyle$ $=$ $\displaystyle \int \frac {\mathrm d u} {u^2}$ simplifying $\displaystyle$ $=$ $\displaystyle \frac {-1} u + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {-1} {\tan \dfrac x 2} + C$ substituting for $u$ $\displaystyle$ $=$ $\displaystyle -\cot \frac x 2 + C$ Cotangent is Reciprocal of Tangent $\displaystyle \implies \ \$ $\displaystyle \int \frac {\mathrm d x} {1 - \cos a x}$ $=$ $\displaystyle \frac {-1} a \cot \frac {a x} 2 + C$ Primitive of Function of Constant Multiple

$\blacksquare$