Find the integral of the function $\frac{\cos 2x + 2\sin^2 x}{\cos^2 x}$.

Given the integral: $\int \frac{\cos 2x + 2\sin^2 x}{\cos^2 x} dx$
Using the trigonometric identity $\cos 2x = 1 - 2\sin^2 x$,we can rewrite the numerator:
$\cos 2x + 2\sin^2 x = (1 - 2\sin^2 x) + 2\sin^2 x = 1$
Substituting this into the integral:
$\int \frac{1}{\cos^2 x} dx$
Since $\frac{1}{\cos^2 x} = \sec^2 x$:
$\int \sec^2 x dx = \tan x + C$
where $C$ is an arbitrary constant.