Slightly Harder Cauchy Integral

[knowlewj01]

Homework Statement

Evaluate the integral $I_1 = \int_0^{2\pi} \frac{d\theta}{(5-3sin\theta)^2}$

Homework Equations

The Attempt at a Solution

I start off by switching the sine term for a complex exponential $e^{i\theta}=cos\theta +isin\theta$
I will consider only the Imaginary component of the solution.

now make the substitution: $z=e^{i\theta}$

so we have:

$I_1 = Im\left(I_2\right)$

$I_2 = \int_0^{2\pi} \frac{1}{(5-3z)^2}\frac{dz}{iz}=\frac{1}{i}\int_0^{2\pi}\frac{dz}{z(5-3z)^2}$

so we have 2 poles: a simple pole at z=0 and a second order pole at z=5/3

I'm not sure wether i need to include both residues. the question does not indicate the contour to integrate. Should it be a unit circle? I'm not sure why it would be. any ideas?

 

[clamtrox]

I'm not sure wether i need to include both residues. the question does not indicate the contour to integrate. Should it be a unit circle? I'm not sure why it would be. any ideas?

The original integral is over θ from 0 to 2π. When you change your integration variable to z=exp(iθ), what values can z take?