Question:

Let $I = \int \frac{\sqrt{\tan x}}{2 \tan x} (1 + \tan^2 x) dx = \int \frac{\sqrt{\tan x}}{2 \tan x} \sec^2 x dx = \int \frac{\sec^2 x dx}{2 \sqrt{\tan x}}$<br>Put $\tan x = t \Rightarrow \sec^2 x = \frac{dt}{dx}$<br>$\therefore I = \frac{1}{2} \int (t)^{-1/2} dt = \frac{t^{1/2} \times 2}{2 \times 1} + C = \sqrt{\tan x} + C$