Answer
$ 2 \pi$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{\pi}^{2\pi} [(-\cos x) +\cos yx)]_{0}^{\pi}=\int_{\pi}^{2\pi}[(1+\pi \cos y) -(-1) dy$
This implies that
$\int_{\pi}^{2\pi} (2+\pi \cos y) dy=(2+\pi \cos y)_{\pi}^{2\pi}$
Hence, $(4 \pi -\pi \sin 2 \pi )-(2 \pi -\pi \sin \pi )=2 \pi$
