Question

Integrate the function $\frac{\sqrt{\tan x}}{\sin x\cos x}$

Answer 1

Let $I=\int \frac{\sqrt{\tan x}}{\sin x\cos x}dx$
$=\int \frac{\sqrt{\tan x}\times \cos x}{\sin x\cos x\times \cos x}dx$
$=\int \frac{\sqrt{\tan x}}{\tan x{\cos }^{2}x}dx$
$=\int \frac{{\sec }^{2}xdx}{\sqrt{\tan x}}$
Put $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$
$\therefore I=\int \frac{dt}{\sqrt{t}}$
$=2\sqrt{t}+C$
$=2\sqrt{\tan x}+C$