Complex integration of real-valued trig function

[brandones]

Homework Statement

Integrate:

$$\int \frac{1}{(3+2cos(θ))} dθ$$ evaluated from zero to pi.

Homework Equations

I can't think of any. All of the integration formulas in the text rely on the existence of a singularity somewhere in the complex plane. This thing is analytic everywhere.

The Attempt at a Solution

Since it's analytic everywhere, its integral over any closed loop equals zero. That tells me that its integral along the x-axis from zero to pi is equal to the negative of its integral along any other path from pi to zero. I tried to find a path along which cos(theta) is linear (which would give it average value zero) to no avail. I have no idea where to go.

 

[grey_earl]

Your integral gives Pi/sqrt(5), calculate the plain ol' indefinite integral and plug in theta=Pi and theta=0.

If you have to do it using complex tricks, I suggest a variable change t = theta-pi/2, which gives $$\int_{-\pi/2}^{\pi/2} \frac{1}{3 - 2 \sin t} \mathrm{d} t$$, and then you can stretch your integration interval using u = tan t. This gives you an integral from -infty to +infty which has singularities for I am u \neq 0. I'm sure you know how to do this then, if not, just ask again.