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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calculus
complex-analysis
trigonometry
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0Here is a related [problem](http://math.stackexchange.com/questions/211058/evaluating-frac12-pi-int-02-pi-frac11-2t-cos-theta-t2d-theta/211068#211068). – 2012-11-28
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0This 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. – 2012-11-28
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1Possible duplicate of [Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$](https://math.stackexchange.com/questions/1061705/evaluate-int-02-pi-frac-sin2-theta54-cos-theta-mathrm-d-theta) – 2017-10-03
1 Answers
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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.
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0$$\sin\theta=\frac{i}{2}\left(e^{i\theta}-e^{-i\theta}\right),$$ And then plug cos and sin into the integrand? – 2012-11-28
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0Or, do I just switch the sin2 theta into 1-cos2 theta and work from there? – 2012-11-28
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0either way works, though the formula you wrote for $\sin\theta$ is off by an overall minus sign. – 2012-11-28