Line integral in complex plane

[juphuy]

In the complex plane, let C be the circle |z| = 2 with positive (counterclockwise) orientation. Then

a. 0
b. 2(pi)(i)
c. (pi)(i)/2
d. (pi)(i)/8
e. (pi)(i)/16

Answer: D

I don't know how to solve it, really no idea . May you help me????

[acd]

Inside the disk |z|=2 , the function has only one pole: z=1. So, with the residue theorem, the integral is 2(pi)(i)Res[f(1)]. Res[f(1)]= 1/16, so the integral is (pi)(i)/8 .