I'm attempting to do a complex contour integral as follows:

Integral along C given as the circle |z|=3 in the CCW (positive) direction. Evaluate integral along C of z^-1 + z^-3 dz directly by parameterization of C.

I parameterized:

z(Ɵ)=3e^iƟ 0<=Ɵ<=2*pi

Since integral of f(z)dz along c is integral from a to b of f(z(Ɵ))z'(Ɵ)dƟ by (reasoning? also, do I need toshowthis is piecewise continuous or is it already taken care of?)

I then integrate [1/(3e^iƟ)+1/((3e^iƟ)^3)]3ie^iƟ dƟ

=integral from 0 to 2pi i+i/9e^2iƟ dƟ

iƟ + -1/18*i*e^-2iƟ

which when evaluated from 0 to 2pi gives me

[2pi*i - 1/18 * i * 1] - [0- 1/18*i]

= 2 * i * pi for my final answer.

(since e^-2pi*i = e^0= 1)

My question, then, is what logical/reasoning steps have I left out and did I get this right? These integrals not done with the residue method really mess me up.

Thanks in advance.