Question

Integrate $\int \frac{1}{1-\cos x}dx$

Answer 1

$\int \frac{1}{1-\cos x}dx=\int \frac{1}{2{\sin }^{2}x/2}dx=\frac{1}{2}\int {\csc }^{2}x/2dx\frac{x}{2}=t;dx=2dt$

$\frac{1}{2}\int {\csc }^{2}t\times 2dt$

$=\frac{1}{2}(-\cot t)=\frac{-1}{2}\left(\cot \frac{x}{2}\right)$