Question

Solve
$\int \left(\frac{x-\sin x}{1-\cos x}\right)dx$.

Answer 1

$\begin{array}{c}\int \frac{x-\sin x}{1-\cos x}dx=\int \frac{x-\sin x}{2{\sin }^{2}x/2}\\ \Rightarrow \int \frac{1x}{2{\sin }^2\frac{x}{2}}-\int \frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{2{\sin }^2\frac{x}{2}}dx\\ \Rightarrow \frac{1}{2}\int x\cos e{c}^2\frac{x}{2}-\int \cot +\frac{x}{2}dx\\ \frac{1}{2}\left(x\times \int \cos e{c}^2\frac{x}{2}dx-\int \cos e{c}^2\frac{x}{2}dx\right)-\int \cot \frac{x}{2}dx\\ \Rightarrow \frac{1}{2}\left(x\times -2\cot \frac{x}{2}\right)-\int -\cot \frac{x}{2}dx-\int \cot \frac{x}{2}dx\\ =-x\cot \left(\frac{x}{2}\right)\end{array}$