Integration by parts

Integration by parts is a trick used to solve integrals by transforming the integral to a less complex one.

Proposition
\[ \int_a^b f(x)g^\prime(x)\mathrm{d}x= \left[f(x)g(x)\right]_a^b - \int_a^b f^\prime(x)g(x)\mathrm{d}x \]
Proof
Problem I
Solve \[ \int_0^{\frac{1}{2}\pi} x\cos x dx \]
Solution