Complex Analysis - Contour integration

Solution Preview

Please see the attachment for detailed solution.

The integral is:
(1.1)
The integration is over the real axis.
Note that
(1.2)
So:
(1.3)
We can write it as a complex integral:
(1.4)
Or:
(1.5)
Where
(1.6)

In order to take advantage of Cauchy's residue theorem, we would like to find a closed path that includes the real axis and some path that we know the value of the integral on.
Let's look at the first integral
If the real axis is the path and the other yet unknown path is then the closed path is and:
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