Question

\int \:xcosxdx

Answer 1

Use integration by parts:

$\int u\:dv=uv-\int v\:du$

In this task $u=x$ and $\cos(x)dx=dv =>du=dx, v=\sin(x)$. So the answer is

$\int x\cos(x)\:dx=x\sin(x)-\int \sin(x)\:dx=x\sin(x)+\cos(x)+const$