Question

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

Answer 1

We use the identity $\cos 2x={\cos }^{2}x-{\sin }^{2}x$

With the substitution of this expression into the given integral, we get:

$\int \frac{\cos 2x+2{\sin }^{2}x}{{\cos }^{2}x}dx=\int \frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}dx=\int \frac{1}{{\cos }^{2}x}dx=\int {\sec }^{2}xdx=\tan x+C$

where $C$ is a real constant of integration.