Question

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

Answer 1

Consider the given integral.

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

$I=\int \frac{\sqrt{\tan x}}{\tan x\cos x\frac{1}{\sec x}}dx$

$I=\int \frac{{\sec }^{2}x\sqrt{\tan x}}{\tan x}dx$

$I=\int \frac{{\sec }^{2}x}{\sqrt{\tan x}}dx$

Let $t=\tan x$

$dt={\sec }^{2}xdx$

Therefore,

$I=\int \frac{1}{\sqrt{t}}dt$

$I=2\sqrt{t}+C$

On putting the value of $t$, we get

$I=2\sqrt{\tan x}+C$

Hence, this is the answer.