contour integral, limiting contour theorem with residue

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.