Question

$\int \frac{\sqrt{\tan x}}{\sin x\cdot \cos x}dx$ is equal to

• A. $\sqrt{\cot x}+C$
• B. $2\sqrt{\tan x}+C$
• C. $\frac{1}{2}\sqrt{\cot x}+C$
• D. $\ln |\sec x-\tan x|+C$

Answer 1

The correct option is B $2\sqrt{\tan x}+C$

$I=\int \frac{\sqrt{\tan x}}{\sin x\cdot \cos x}dx=\int \frac{{\sec }^{2}x}{\sqrt{\tan x}}dx$
Putting $\tan x=t$
$\Rightarrow {\sec }^{2}xdx=dt$
$\therefore I=\int \frac{1}{\sqrt{t}}dt=2\sqrt{t}+C=2\sqrt{\tan x}+C$