Chapter 4 - Integration - 4.2 The Indefinite Integral - Exercises Set 4.2 - Page 279: 32

$$\frac{1}{2}\tan x +c$$

Given$$\int \frac{dx}{1+\cos 2x}$$ Since $$\cos 2x=2\cos^2x-1$$ \begin{align*} \int \frac{dx}{1+\cos 2x} &=\int \frac{dx}{2\cos^2x}\\ &=\frac{1}{2}\int \sec^2 xdx\\ &=\frac{1}{2}\tan x +c \end{align*}