Explanation

(d)
$\int_{0}^{\pi/4}{(\cos{x - \sin{x)dx + \int_{\pi/4}^{5\pi/4}{(\sin{x - \cos{x)dx} + \int_{2\pi}^{\pi/4}{(\cos{x - \sin{x)dx}}}}}}}}$
$= \lbrack\lbrack{\sin{x + \cos{x\rbrack}}}_{0}^{\pi/4} - \lbrack{\sin{x + \cos{x\rbrack}}}_{\pi/4}^{5\pi/4} + \left\lbrack {\sin{x + \cos{x\rbrack}}}_{2\pi}^{\pi/4} \right\rbrack$
$= \left\lbrack \sin{\frac{\pi}{4} + \cos{\frac{\pi}{4} - (\sin{0 + \cos{0)}}}} \right\rbrack - \left\lbrack \sin{\frac{5\pi}{4} + \cos{\frac{5\pi}{4} - \left( \sin{\frac{\pi}{4} + \cos\frac{\pi}{4}} \right)}} \right\rbrack$
$+ \left\lbrack \sin{\frac{\pi}{4} + \cos{\frac{\pi}{4} - (\sin{2\pi + \cos{2\pi)}}}} \right\rbrack$
$= \left\lbrack \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1 \right\rbrack - \left\lbrack - \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} - \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \right) \right\rbrack + \left\lbrack \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1 \right\rbrack$
$= \left\lbrack \sqrt{2} - 1 \right\rbrack - \left\lbrack - \sqrt{2} - \sqrt{2} \right\rbrack + \left\lbrack \sqrt{2} - 1 \right\rbrack$
$= \left\lbrack \sqrt{2} - 1 + 2\sqrt{2} + \sqrt{2} - 1 \right\rbrack = 4\sqrt{2} - 2$