Explanation

(a)
$\int_{}^{}{\left( \sqrt{\tan x} + \sqrt{\cot x} \right)dx}$
$= \int_{}^{}\left( \frac{\sqrt{\sin x}}{\sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x}}\ \right)dx$
$= \int_{}^{}{\left( \frac{\sin{x + \cos x}}{\sqrt{\sin{x\cos x}}} \right)dx}$
$= \int_{}^{}{\frac{\sqrt{2}(\sin{x + \cos x})}{\sqrt{2\sin{x\cos x}}}dx}$
$= \sqrt{2}\int_{}^{}\frac{\sin{x + \cos x}}{\sqrt{1 - (\sin{x - {\cos{x)}}^{2}}}}$
Put $\sin{x - \cos{x = t}}$
$\Rightarrow (\cos{x + \sin{x)dx = dt}}$
$= \sqrt{2}\int_{}^{}\frac{dt}{\sqrt{1 - t^{2}}}$
$= \sqrt{2}\sin^{- 1}{(\sin{x - \cos{x) + c}}}$