You can appeal to the Cauchy-Goursat Theorem which says that if f(z) is analytic inside and on a closed contour C then . Alternatively you can easily modify the argument given in case 2: n < 0.
Case 2: n < 0.
Let , that is, the circle with radius and centre at . On :
(that is, ): .
Case 3: n = 0.
Left for you to do.
These results readily follow from Cauchy's Integral Theorem (which you might not have met yet).
It follows from the Cauchy-Goursat Theorem that when the circle does not contain the origin, the integral will equal zero. Alternatively, the case 2 argument is readily modified.