Contour Integral of sin θ, cos θ

Question

How to solve this question using contour integration

$$\int_0^{2π} \frac {sin^2 θ }{a + b cosθ} dθ $$

The problem what I am facing here is due to square of sinθ , it is getting very complicated and answer is not coming. Please help me out.

Answer 1

Use $z=e^{i\theta}$ then \begin{align} I &=\int_{0}^{2\pi} \frac{\sin^2\theta}{a+b\cos\theta}\, d\theta\\ &=\frac12\int_{0}^{2\pi} \frac{1-\cos2\theta}{a+b\cos\theta}\, d\theta\\ &=\frac12\int_{|z|=1} \frac{{\bf Re\,}(1-z^2)}{a+b(z+z^{-1})}\,\dfrac{1}{iz} dz,\\ &={\bf Re\,}\dfrac{1}{2i}\int_{|z|=1} \frac{1-z^2}{bz^2+az+b}\, dz \end{align}