Explanation

(d)
Let $I = \sqrt{2}\int_{}^{}\frac{\sin x\ }{\sin\left( x - \frac{\pi}{4} \right)}dx$
Put $x - \frac{\pi}{4} = t \Rightarrow dx = dt$
$\therefore\ I = \sqrt{2}\int_{}^{}\frac{\sin\left( \frac{\pi}{4} + t \right)\ dt}{\sin t}$
$= \sqrt{2}\int_{}^{}\frac{\sin\frac{\pi}{4}\cos t + \cos\frac{\pi}{4}\sin t\ }{\sin t}\ dt$
$= \sqrt{2}\int_{}^{}\left( \frac{1}{\sqrt{2}}\frac{\cos t}{\sin t} + \frac{1}{\sqrt{2}} \right)\ dt$
$= \int_{}^{}{(\cot{t + 1})}dt$
$= \log{|\sin t|} + t + c_{1}$
$= x + \log\left| \sin\left( x - \frac{\pi}{4} \right) \right| + c,\ \left( \because c_{1} - \frac{\pi}{4} = c \right)$