Partial fractions is good.
The first integral on the RHS is zero, because the integrand is analytic inside and on the entire contour. Just invoke the Cauchy-Goursat theorem.
For the second integral, there are multiple ways to do it. The easiest is residues, but if you haven't studied that yet, you could do a parametrization. However, I wouldn't pick your parametrization. The first thing I would do is a change of variables: Let be the unit circle centered at the origin. Then
Can you continue from here?