Dear Sang Won Lee ,

First, you need to calculate the imaginary part of your function.

Then the integral can be calculated by two simple changes of variable and using the known derivative of arctanx, which is 1/(x2+1).

Also, you need to use that arctan(infinity)=π/2 and arctan(-infinity)=-π/2.

It is not necessary to perform contour integration.

All details are in the attached pdf file.

Dr Spiros Konstantogiannis excellent solution

I have a lot of examples in my book, Complex Variables, which will be free on 1/4/22. The examples and software are always free. https://www.amazon.com/dp/B07MW6782P

Prof. Maged G. Bin-Saad , thank you very much.