Evaluate$ \ \int_0^{2 \pi} \frac{\sin^2 \theta}{5 + 4 \cos \theta}\,d \theta \ $ using contour integration and the calculus of residues

Question

Evaluate$ \ \int_0^{2 \pi} \frac{\sin^2 \theta}{5 + 4 \cos \theta}\,d \theta \ $ using contour integration and the calculus of residues

Answer 1

Put $z=e^{i\theta}$ so that you're integrating counter-clockwise around the unit circle in the complex plane. Express your integrand in terms of $z$, using $$\cos\theta=\frac{1}{2}\left(e^{i\theta}+e^{-i\theta}\right),$$ etc. Then it should be a straightforward residue problem.