Question

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

Answer 1

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

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

Putting $\tan x=t$

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

$I=\int \sqrt{t}dt=\frac{2}{3}{t}^{3/2}+c$

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