Question

Find the integrals (primitives):
$\rm \displaystyle\int \frac{1}{1-\cos \frac{x}{2} }\ dx$

Answer 1

We know that from half angle formula

$1 - cos \frac{x}{2} = 2 {sin}^{2} (\frac{x}{4} )$
$\int \frac{1}{1-\cos \frac{x}{2} }\ dx \\ \\ \int \frac{1}{2 \: {sin}^{2} \frac{x}{4} }\ dx \\ \\ = \frac{1}{2} \int {cosec}^{2} \frac{x}{4} \: dx$
we know that
$\int \: {cosec}^{2} x \: dx = - cot \: x + C$
So
$\frac{1}{2} \int {cosec}^{2} \frac{x}{4} \: dx \\ \\ = \frac{4}{2}( - cot \: \frac{x}{4} ) + C \\ \\ \int \frac{1}{1-\cos \frac{x}{2} }\ dx= - 2cot \: \frac{x}{4} + C$
Hope it helps you.