Question

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

Answer 1

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

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

$=\int \frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}dx=\int \frac{1}{{\cos }^{2}x}dx$

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

$=\tan x+C$