You need to compute this:
This is the part which I'm not really sure. I'm guessing the is outside because it's less than unity, but I start having trouble when it comes to identifying more complicated poles. What is the right way?
The answer is not a negative, so it may be a careless mistake, but I've pored through the equations without spotting an error.
PS: Is \\ the tag for line break? I can't seem to get it to work.
You pick z = 1/6, because that pole is inside the contour, and the other one is not. Only poles inside a contour have any effect whatsoever on the value of the integral. The implied contour over which you're integrating is the unit circle. So the correct poles at which to evaluate the residues are the ones inside that contour. Make sense?
Do you know what the contour is? Whether the integrand is quadratic or not has nothing to do with the contour.
The original problem was to integrate
which you changed into the contour integral . How did you do that?
What was "z" in terms of " ". As goes from 0 to , how does z change?