Question

Evaluate: $I=\int x\sin 2xdx$

Answer 1

$I=\int x\sin 2xdx$

Let $U=x$; Let $V^{\prime }=\sin 2x$

$U^{\prime }=1$; $V=\frac{-\cos 2x}{2}$

$\therefore \int x\sin 2xdx$

$UV-\int U^{\prime }V$

$=\frac{-1}{2}x\cos 2x-1\cdot \frac{-\cos 2x}{2}$

$=\frac{-1}{2}x\cos 2x+\frac{1}{2}\int \cos 2x$

$=\frac{-1}{2}x\cos 2x+\frac{1}{4}\sin 2x+c$.