Nevermind. I found it. I had forgotten about the coefficient on z^2 when I wrote the denominator in factored form which brings an additional factor of 3 into the denominator.
I need to calculate the definite integral of dx/(4+3cos(x)) from 0 to 2pi.
I believe that the integral from 0 to 2pi of (f(e^ix))*i*(e^ix)dx is equivalent to the integral over the unit circle of f(z)dz.
If that's true, then this problem boils down to finding f(z) so that...