Question

Evaluate
$\int x.{\sin }^{2}xdx$

Answer 1

solution :

$\to \int xsi{n}^{2}xdx$

$\int si{n}^{2}xdx=\int \frac{1}{2}(1-cos2x)dx=\frac{1}{2}(x-\frac{sin2x}{2})+c$

we take $u=x_1$ $v=si{n}^{2}x$

$u^{\prime }=1$ $v^{\prime }=\frac{1}{2}(x-\frac{sin2x}{2})$

w.r.t $\int uv=uv-\int u^{\prime }v$

so $\int xsi{n}^{2}xdx=\frac{x}{2}(x-\frac{sin\left(2x\right)}{2})-\int \frac{1}{2}(x-\frac{sin2x}{2})$

$=\frac{x}{2}(x-\frac{sin2x}{2})-\frac{1}{2}(x^2+\frac{co{s}^{2}x}{2})$

on simpiflying we get

Ans $=\frac{x^2}{4}-x\frac{sin\left(2x\right)}{4}-\frac{cos\left(2x\right)}{8}$