Question

Solve $\int x{\sin }^{2}xdx$

Answer 1

$\int x{\sin }^{2}xdx(\because \cos 2x=1-2{\sin }^{2}x)$

$\int x\left(\frac{1-\cos 2x}{2}\right)dx$

$=\frac{1}{2}\int (x-x\cos 2x)dx$

$=\frac{x^2}{4}-\frac{1}{2}[\frac{x\sin 2x}{2}-\int \frac{\sin 2x}{2}dx]$

$=\frac{x^2}{4}-\frac{x}{4}\sin 2x-\frac{\cos 2x}{8}+c$