Question:

Let $I = \int_{0}^{\pi/2} \frac{x + \sin x}{1 + \cos x} dx = \int_{0}^{\pi/2} \frac{x + 2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^2 \frac{x}{2}} dx$<br>$= \int_{0}^{\pi/2} \left( \frac{x}{2} \sec^2 \frac{x}{2} + \tan \frac{x}{2} \right) dx = \left[ x \tan \frac{x}{2} \right]_{0}^{\pi/2}$<br>$= \frac{\pi}{2}$