Question

Evaluate ∫ x cos 3x dx.

Answer 1

This can be integrated using the method "integration by parts." The formula is
$\int{u\,dv}=uv-\int{v\,du}$

Here, it is convenient to assign
$u=x\\v=\frac{1}{3}\sin{(3x)}$

Then you have
$\int{(x\cos{3x})}\,dx=\frac{1}{3}x\sin{(3x)}-\int{(\frac{1}{3}\sin{(3x))}}\,dx\\\\=\frac{1}{3}x\sin{(3x)}+\frac{1}{9}\cos{(3x)}$