Question

$\int \frac{\cos 2x+2{\sin }^{2}x}{{\cos }^{2}x}dx$

Answer 1

Consider the following question.

$I={\int }^{}\frac{\cos 2x+2{\sin }^{2}x}{{\cos }^{2}x}dx$

$={\int }^{}\frac{{\cos }^{2}x-{\sin }^{2}x+2{\sin }^{2}x}{{\cos }^{2}x}dx$

$={\int }^{}\frac{{\sin }^{2}x+{\cos }^{2}x}{{\cos }^{2}x}dx$

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

$=\tan x+C$

Hence, this is the required answer.