Explanation

(d)
$\int_{0}^{\pi}{|\cos{x|dx = \int_{0}^{\pi/2}{\cos{x\ dx - \int_{\pi/2}^{\pi}{\cos{x\ dx}}}}}}$
$= \left\lbrack \sin x \right\rbrack_{0}^{\pi/2} - \left\lbrack \sin x \right\rbrack_{\pi/2}^{\pi}$
$= \sin{\frac{\pi}{2} - \sin{0 - \sin{\pi + \sin\frac{\pi}{2}}}}$
$= 2\sin{\frac{\pi}{2} = 2}$