How to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$?

Question

I wish to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$.

I guess that I should somehow apply Cauchy's integral formula here, but $\frac{1}{z},\frac{1}{z-1}$ are not analytical inside of $C$ so I can't define something like $f(z)=\frac{1}{z(z-2)}$ and calculate $\oint_{C}\frac{f(z)dz}{(z-1)}$ by Cauchy's.

Can someone please help me understand how to calculate this integral ?

I am guessing there is some trick so I can use Cauchy's integral formula, but I didn't manage to think of any such tricks.

Answer 1

$$ \oint_c \frac{1}{z(z-1)(z-2)} dz = 2\pi i \; \left [ \; \text{Res}\left (f(z), 0\right ) + \text{Res}\left ( f(z), 1 \right ) \; \right ] = 2\pi i(1/2 - 1) = -\pi i $$

Or simply, since pole of $z=2 > 1.5 $

$$ \oint_c \frac{1}{z(z-1)(z-2)} dz = \oint_c \left( \frac{1}{2 (-2+z)}-\frac{1}{-1+z}+\frac{1}{2 z} \right ) dz \\ = \oint_c \frac{1}{2z}dz - \oint_c\frac{1}{z-1} = \pi i - 2 \pi i = -\pi i$$

I hope I am not wrong!!