Explanation

(c)
$\int_{}^{}\frac{dx}{\sin{x - \cos{x + \sqrt{2}}}}$
$= \int_{}^{}\frac{dx}{\sin{x\frac{\sqrt{2}}{\sqrt{2}} - \cos{x\frac{\sqrt{2}}{\sqrt{2}} + \sqrt{2}}}}$
$= \int_{}^{}\frac{dx}{\sqrt{2}(\sin{x\sin{\frac{\pi}{4} - \cos{x\cos{\frac{\pi}{4} + 1)\ }}}}}$
$= \frac{1}{\sqrt{2}}\int_{}^{}\frac{dx}{1 - \cos\left( \ x + \frac{\pi}{4}\ \right)}$
$= \frac{1}{\sqrt{2}}\int_{}^{}\frac{dx}{1 - {cos\ 2}\left( \frac{x}{2} + \frac{\pi}{8}\ \right)}$
$= \frac{1}{\sqrt{2}}\int_{}^{}\frac{dx}{2\sin^{2}\left( \frac{x}{2} + \frac{\pi}{8}\ \right)}$
$= \frac{1}{2\sqrt{2}}\int_{}^{}{cosec^{2}\left( \frac{x}{2} + \frac{\pi}{8}\ \right)dx}$
$= - \frac{1}{2\sqrt{2}} \bullet \frac{- \cot\left( \frac{x}{2} + \frac{\pi}{8}\ \right)}{\frac{1}{2}}$+c=$\frac{1}{\sqrt{2}}\cot{\left( \frac{x}{2} + \frac{\pi}{8}\ \right) + c}$