By observing that the function $|\cos|$ is $2\pi$-periodic, calculate, for $\omega \in \mathbf{R}$, the integral
$$\int_{-\pi}^{\pi} |\cos(u - \omega)| \mathrm{d}u$$
Deduce that, if $(a,b) \in \mathbf{R}^{2}$,
$$\int_{-\pi}^{\pi} |a\cos(u) + b\sin(u)| \mathrm{d}u = 4\sqrt{a^{2} + b^{2}}$$