Evaluate$ \ \int_0^{2 \pi} \frac{\sin^2 \theta}{5 + 4 \cos \theta}\,d \theta \ $ using contour integration and the calculus of residues
Asked
Active
Viewed 1,754 times
-1
-
Here is a related problem. – Mhenni Benghorbal Nov 28 '12 at 09:01
-
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
-
1Possible duplicate of Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta},\mathrm d\theta$ – kingW3 Oct 03 '17 at 11:25
1 Answers
0
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.
Jonathan
- 8,358
-
$$\sin\theta=\frac{i}{2}\left(e^{i\theta}-e^{-i\theta}\right),$$ And then plug cos and sin into the integrand? – Angus Leo Nov 28 '12 at 06:32
-
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