Hi, can anyone please help me with this question, thanks a lot.

I did try to take the contour, and take notice the three "bad points" are

,

, and

.

I used residue theorem that

.

I can use limiting contour theorem to get one integral is

.

However, I'm really having trouble solve this question, I thought my methods are right, but I can't get the right answer which is

. One friend told me I need to worry about choosing branch because of that

, but I don't quite understand it and what I supposed to do.

Can anyone please show me some precise steps on solving this question? Thanks a lot.