This implies that
Since we enclosed the pole at we know the radius of the arc
This gives us the inequality
This gives us an upper bound on the valus of the integrand. So we know the value of the integral cannot be larger than its maximum multiplied by the lenth of the curve of integration.
So the integral looks like
Becuase of the bound we know that the 2nd integral goes to zero as R goes to infitity so we get
As far as the other pole goes you don't need it
The residue theorem states the values of an integral on any closed curve is the sum of the residues inside the curve. The poles outside do not affect the value of the integral.
I hope this helps.