Question

Evaluate the following:
$\int x{\sin }^{2}x.dx$ ?

Answer 1

We start by integrating the function${sin}^2\left(x\right)$

so $\int {sin}^2\left(x\right)dx=\int \frac{1}{2}(1-cos(2x)=\frac{1}{2}(x-sin{\left(2x\right)}^2)+C$

so then you do the original integral by party.you take:

$u=x$

$u=1$

$v={sin}^2\left(x\right)=\frac{1}{2}(x-sin{\left(2x\right)}^2)$

we know that $\int u.v=uv-\int uv$

$\int {sin}^2\left(x\right)dx=x^2(x-sin{\left(2x\right)}^2-\int \frac{1}{2}(x-sin{\left(2x\right)}^2)=\frac{x}{2}\left(\frac{x-sin\left(2x\right)}{2}\right)-\frac{1}{2}(x^2+\frac{cos\left(2x\right)}{4})$

Which then can be simplified to

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