For the function

$$f(z)=\frac{1}{z(z-2)}$$

find the Laurent expansions

(i) about $z=0$ in the annulus $0<|z|<2$,

(ii) about $z=0$ in the annulus $2<|z|<\infty$,

(iii) about $z=1$ in the annulus $0<|z-1|<1$.

What is the nature of the singularity of $f$, if any, at $z=0, z=\infty$ and $z=1$ ?

Using an integral of $f$, or otherwise, evaluate

$$\int_{0}^{2 \pi} \frac{2-\cos \theta}{5-4 \cos \theta} d \theta$$