Help with path-Integral (Complex Analysis)

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I seem to be stuck on one path-integral and want to make sure how to solve it properly. As I have seen a lot of different solutions, maybe someone can give me some clarity

The Question

Part (a) is clear. Here I do not seem to have any problems. However after that, I get into some troubles solving the first Integral. This is what I tried. I am not sure if I did it totally wrong

My attempt