Explanation

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