Question

Integrate the function $\frac{{\sec }^{2}x}{\sqrt{{\tan }^{2}x+4}}$

Answer 1

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