Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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
up vote 0 down vote accepted

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.

share|cite|improve this answer
$$\sin\theta=\frac{i}{2}\left(e^{i\theta}-e^{-i\theta}\right),$$ 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.