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

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

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 Here is a related problem. – Mhenni Benghorbal Nov 28 '12 at 9:01 This is your third problem in a very little while. The way you ask questions is not considered polite in this site. Please refer to FAQ about this, and anyway: it'd be refreshing and nice to see some self work, ideas from you on these problems. – DonAntonio Nov 28 '12 at 12:58

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.
 $$\sin\theta=\frac{i}{2}\left(e^{i\theta}-e^{-i\theta}\right),$$ And then plug cos and sin into the integrand? – Angus Leo Nov 28 '12 at 6:32 Or, do I just switch the sin2 theta into 1-cos2 theta and work from there? – Angus Leo Nov 28 '12 at 18:00 either way works, though the formula you wrote for $\sin\theta$ is off by an overall minus sign. – Jonathan Nov 28 '12 at 19:03