Question

Evaluate: $\int \frac{x}{1-\cos x}\ dx$

Answer 1

Solution:

As we know that

$1- cos x = 2 sin^{2}(\frac{x}{2})$

Put this value in the given function to convert in integrable form

$\int\frac{x}{2 sin^{2}\frac{x}{2}}dx\\\\=\frac{1}{2}\int x\:cosec ^{2}(\frac{x}{2})dx$

Now integrate by parts

$\int x\:cosec ^{2}(\frac{x}{2})dx=x\int cosec ^{2}(\frac{x}{2})dx -\int[ \frac{dx}{dx}\int cosec ^{2}(\frac{x}{2})dx]dx\\\\=-2x cot(\frac{x}{2})+2\int cot(\frac{x}{2})dx\\\\\int \frac{x}{1-\cos x}\ dx= -2x cot(\frac{x}{2})+4\: log\:sin(\frac{x}{2})+C$