Question

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

Answer 1

Now

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

$=\int \frac{\sqrt{\tan x}.{\sec }^{2}x}{\tan x}dx$ [ Dividing the numerator and the denominator by ${\cos }^{2}x$]

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

$=\int \frac{d\left(\tan x\right)}{\sqrt{\tan x}}$

$=2\sqrt{\tan x}+c.$ [Where $c$ is integrating constant]