\begin{align} \int_0^{2 \pi} \frac{\cos \theta}{2 + \cos \theta} d \theta &= 2 \int_0^\pi \frac{\cos \theta}{2 + \cos \theta} d \theta \\ &= 2 \int_0^\infty \frac{\frac{1-t^2}{1+t^2}}{2 + \frac{1-t^2}{1+t^2}} \frac{2 dt}{1 + t^2} \ \ \ \ \ \ \ \ \left( t = \tan \frac{\theta}{2} \right) \\ &= -4 \int_0^\infty \frac{t^2 - 1}{(t^2 + 3)(t^2 + 1)} dt \\ &= -4 \int_0^\infty \left( \frac{2}{t^2 + 3} - \frac{1}{t^2 + 1} \right) dt \\ &= -4 \left( \frac{\pi}{\sqrt{3}} - \frac{\pi}{2} \right) \\ &= - \frac{4 \pi}{\sqrt{3}} + 2 \pi \end{align}