Question

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

Answer 1

$$\begin{gathered} \int \frac{{\sec }^{2}xdx}{\sqrt{4+{\tan }^{2}x}} \\ \mathrm{let}\tan x=t \\ \Rightarrow {\sec }^{2}xdx=dt \\ \mathrm{Now},\int \frac{{\sec }^{2}xdx}{\sqrt{4+{\tan }^{2}x}} \\ =\int \frac{dt}{\sqrt{2^2+t^2}} \\ =\log \left|t+\sqrt{4+t^2}\right|+C \\ =\log \left|\tan x+\sqrt{4+{\tan }^{2}x}\right|+C \end{gathered}$$