Evaluate: ∫cos2x + 2sin2xcos2x dx

We use the identity cos2x = cos²x - sin²x. With the substitution of this expression into the given integral, we get:
∫cos2x + 2sin2xcos2x dx = ∫cos2x + sin2xcos2x dx = ∫1cos2x dx = ∫sec2x dx = tanx + C
where C is a real constant of integration.