Question

Integrate with respect to $x$:
$x{\sin }^{2}x$

Answer 1

$\int \underset{\left(i\right)}{x}\underset{\left(ii\right)}{{\sin }^{2}x}dx$

$=x\int {\sin }^{2}xdx-\int 1\left\{\int {\sin }^{2}xdx\right\}dx$

Now,

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

$=\frac{1}{2}\int 2{\sin }^{2}xdx$

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

$=\frac{1}{2}\left[x-\frac{\sin 2x}{2}\right]$

$=\frac{x}{2}-\frac{\sin 2x}{4}$

$\int x.{\sin }^{2}xdx=x\times \left(\frac{x}{2}-\frac{\sin 2x}{4}\right)-\int \left(\frac{x}{2}-\frac{\sin 2x}{4}\right)dx$

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

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

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