Hey guys, I was hoping you'd be able to help me with a question. I got the right answer but my method of doing it was a little different to the solution to the answer in my textbook. The question was this:

What I did was use notice that the above integral was the same as:

I then went onto the complex plane and made a semicircle closed contour with infinite radius as shown below:

I found the singularities of the function, with two of them being inside the contour (e^(3pi/4)i and e^(pi/4)i) and then applied Cauchy's Residue Theorem:

Worked out the values of the residues of the two poles inside the contour and came to a final answer of:

What my maths lecturer did was way more in depth and used a lot of weird things that did not make too much sense to me and are too long to present here. Is there anything wrong with the way I solved that integral? Surely if you have a closed contour with a pole in it you can straight away use Cauchy's Residue Theorem to find the value of the residue at the pole and multiply it by 2(pi)i?