Question

Evaluate :

$\int \sqrt{\tan x}(1+{\tan }^{2}x)dx$

Answer 1

$I=\int \sqrt{\tan x}(1+{\tan }^{2}x)dx$

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

Put $\tan x=t$

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

Thus, $I=\int \sqrt{t}dt$

$I=\frac{t^{3/2}}{3/2}+C$

$I=\frac{2}{3}(\tan x{)}^{3/2}+C$