Applications of Residue Theorem

Bonjour,

I'm just learning how to apply the residue theorem and manipulating functions to solve integrals that one previously couldn't solve. Some examples I have come across are:

a)Suppose a, b > 0 and a is not equal to b

b)

c) Suppose a > 0.

I am mainly having trouble with the manipulations. I've tried to understand it through using the textbook and attempting the problems above but I end up with integrals I can't solve. The answers for each question respectively are:

a)

b)

c)

I appreciate all your help in understanding this tough concept. Thanks!

ComplexXavier