Question

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

Answer 1

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

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

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

Let $\tan x=t$

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

Substituting in the integral,

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

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

$=2\sqrt{t}+c$

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