Complex Analysis - Contour Integration

[QuantumJG]

In a lecture today we evaluated a integral:

\oint_{\Gamma} \dfrac{3z - 2}{z^2 - z} dz

Where,

\Gamma = \{ z \in \mathbb{C} | |z| + |z-1| = 3 \}

Our lecturer evaluated it to be 6πi

I sort of understood how he did it, but he really rushed through his steps.

 

[hunt_mat]

So what is you specific question? You have two poles in your integrand z=0 and z=1, you use Cauchy's residue theorem to do the integral and you need to compute residues. However ONLY the poles that are inside the contour are counted.

Can you draw the contour?