Question

Find the anti derivative of $\frac{{\sec }^{2}x}{\sqrt{{\tan }^{2}x+4}}$.

Answer 1

$\int \frac{{\sec }^{2}x}{\sqrt{{\tan }^{2}x+4}}dx$
$\tan x=t$

${\sec }^{2}xdx=dt$
$\int \frac{dt}{\sqrt{t^2+2^2}}$
$=\log |t+\sqrt{t^2+2^2}|$
$=\log |\tan x+\sqrt{{\tan }^{2}x+4}|+C$