Question

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

Answer 1

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

Let $\tan x=z⟹se{c}^{2}xdx=dz$

Therefore,

$I=\int \frac{1}{\sqrt{z^2+4}}dz$

$I=\log |z+\sqrt{z^2+4}|+c$

$I=\log |\tan x+\sqrt{{\tan }^{2}x+4}|+c$

Hence, this is the answer.