Question

Evaluate: $\int x\sin 2xdx$

Answer 1

$I=\int x\sin 2x$

We know the formula (by Parts)

$\int uv=u\int v-\int (\int v)\frac{du}{dn}dx$

$u=x;v\sin 2x$

$I=x\int \sin 2x-\int (\int \sin 2x)\frac{d}{dx}\left(x\right)dx$

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

$I=\frac{-x\cos 2x}{2}+\frac{\sin 2x}{2.2}+C$

$I=\frac{\sin 2x}{4}-\frac{x\cos 2x}{2}+C$.