Question

Solve $\int \frac{\sqrt{\tan x}}{\sin x\cos x}dx$

Answer 1

$\int \frac{\sqrt{\tan x}}{\sin x\cos x}dx$

Put, $\tan x=t$

$⟹{\sec }^{2}xdx=dt$

Now, $\int \frac{\sqrt{\tan x}}{\sin x\cos x}dx=\int \frac{\sqrt{\tan x}}{\tan x{\cos }^{2}x}dx$

$=\int \frac{{\sec }^{2}x}{\sqrt{\tan x}}dx=\int \frac{1}{\sqrt{t}}dt$

$=2\sqrt{t}+c$

$=2\sqrt{\tan x}+c$