Question:

$I = \int \frac{x + \sin x}{1 + \cos x} dx = \int \left( \frac{x + \sin x}{2 \cos^2 \frac{x}{2}} \right) dx = \int \left( \frac{1}{2} x \sec^2 \frac{x}{2} + \tan \frac{x}{2} \right) dx = x \tan \frac{x}{2} + C$