Integrating $\int x\sin x dx$

Question

Could someone outline the step-by-step approach for the following indefinite integral?

$$\int x\sin x \ dx$$

I know the solution is $\sin(x)-x\cos(x)$, but cannot see how one would go about step-wise solving for it in a logical manner that does not include arbitrary guessing (even with intelligent guessing) of terms to use.

Thanks a bunch.

Answer 1

Use Integration by parts

$$\int x \sin x dx = -x \cos x - \int \cos x dx$$