Here I go again...
All of these line! integrals are on the circle |z|=2, with counterclockwise orientation.
Unfortunately I cannot use the residue theorem.
For numbers 4 and 5 I have been trying to write the integrands as power series, but I am having trouble relating those to line integrals.
For numbers 1, 2 and 3 I feel like I have to split them up into a real and imaginary parts and then do two integrals, but I am having trouble doing this.
And throughout I am always wondering why the line integral is evaluated at a circle of radius 2, instead of the usual 1. I think some of the functions have singularities in that region, but I'm not sure what that does.