Answer
$4$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{-\pi}^{0} [(-y \cos (x+y)_{0}^{\pi} +\int_{0}^{\pi} \cos (x+y) dy] dx=\int_{-\pi}^{0} \pi \cos x-\sin x-\sin x dx$
This implies that
$[\pi \sin x+\cos x+\cos x]_{-\pi}^{0}=(\pi \sin 0+\cos 0+\cos 0)-(\pi \sin \pi+\cos \pi+\cos \pi)$
Hence, $1+1+1+1=4$
