The question

Evaluate using the Cauchy Integral Formula, where the circle is traversed once anti-clockwise.

My attempt

z = 3, z = 2 are both singularities in the contour, so:

Rewrite the integrand such that one of the singularities is in the denominator:

Thus by CIF,

This singularity is of pole 2, so we must use :

= 72\pi i

Now for the other singularity:

Thus by CIF,

This singularity is of pole 3, so we must use :

=

Add them together, and we get . However, the answer is actually 0. I think I'm supposed to be getting for the second one, but I can't work out where I've gone wrong. :/

Any help would be greatly appreciated!