Question

Evaluate: $\int x\sin 3xdx$

Answer 1

Given, $\int x\sin 3xdx$

Let $I=\int x\sin 3xdx$

$\Rightarrow$ integration by part

$\int u.vdx=u\int vdx$ $-\int (\frac{du}{dx}\int vdx)dx$

$I=x\int \sin 3x-\int 1\cdot \int \sin 3xdx\cdot dx$

$=\frac{x(-\cos 3x)}{3}+\int \frac{\left(\cos 3x\right)}{3}dx$

$I=-\frac{x}{3}\cos 3x+\frac{\sin 3x}{9}+C$