Explanation

(c)
Let $I = \int_{}^{}\frac{\sqrt{\tan x}}{\sin x\cos x}\ dx$
$= \int_{}^{}\frac{\sqrt{\sin x}}{{\sqrt{\cos x}\sin}x\cos x}\ dx$
$= \int_{}^{}\frac{1}{\sqrt{\sin x}\cos^{\frac{3}{2}}x}\ dx$
$= \int_{}^{}\frac{1}{\sqrt{\tan x}\cos^{2}x}\ dx$
$= \int_{}^{}\frac{\sec^{2}x}{\sqrt{\tan x}}\ dx$
Put $\tan x = t \Rightarrow \sec^{2}x\ dx = dt$
$\therefore\ I = \int_{}^{}\frac{1}{\sqrt{t}}\ dt = 2\sqrt{t} = 2\sqrt{\tan x}$