Prove

Let $\begin{aligned} I &=\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x \\ &=\int \frac{\sqrt{\tan x} \times \cos x}{\sin x \cos x \times \cos x} d x \\ &=\int \frac{\sqrt{\tan x}}{\tan x \cos ^{2} x} d x \\ &=\int \frac{\sec ^{2} x d x}{\sqrt{\tan x}} \end{aligned}$

Let $\tan x=t \Rightarrow \sec ^{2} x d x=d t$

$\begin{aligned} \therefore I &=\int \frac{d t}{\sqrt{t}} \\ &=2 \sqrt{t}+\mathrm{C} \\ &=2 \sqrt{\tan x}+\mathrm{C} \end{aligned}$