Solve $\int_{|z|=3}\frac{2z^2-z+1}{(z-1)^2(z-2)}dz$

Question

Find $$\int_{|z|=3}\frac{2z^2-z+1}{(z-1)^2(z-2)}dz$$

I thought of using partial fractions so i ended up with

$$\int_{|z|=3}\frac{-5}{z-1}+\frac{-2}{(z-1)^2}+\frac{7}{z-2} dz$$

I did this because both singularities at 1 and 2 matter. After this i would apply the cauchy integral formula to get the solution. is that right?

Answer 1

Yes, that is right.

Another way of solving the problem consists in applying the residue theorem. In order to do so, you would have to compute the residue of $\frac{2z^2-z-1}{(z-1)^2(z-2)}$ at the points $1$ and $2$.