Question

$\int \frac{\sqrt{\tan x}dx}{\sin x\cos x}=k\sqrt{\tan x}.$ Find the value of $k$.

Answer 1

$I=\int \frac{\sqrt{\tan x}}{\sin x\cos x}dx$
$=\int \sqrt{\tan x}\csc x\sec xdx$
$=\int \frac{{\sec }^{2}x}{\sqrt{\tan x}}dx$
Substitute $\upsilon =\tan x\Rightarrow d\upsilon =\upsilon {\sec }^{2}xdx$
$\therefore I=\int \frac{1}{\sqrt{\upsilon }}d\upsilon =2\sqrt{\upsilon }=2\sqrt{\tan x}.$