Exercise: Evaluating integration $\int_{|z|=r} \frac{1}{(z-a)(z-b)}dz$, $|a|<r<|b|$

why don't you expand $$\frac{1}{z - a} = \frac{1}{z} \frac{1}{1 - a/z} = \frac{1}{z}\{1 + a/z + a^2/z^2 + \cdots \}$$ and $$\frac{1}{z - b} = -\frac{1}{b} \frac{1}{1 - z/b} = -\frac{1}{b}\{1 + z/b + z^2/b^2 + \cdots \}$$ and use the fact $\int_{|z| = r} z^n dz = 0$ for $n \neq - 1$ and $\int_{|z| = r} z^{-1} dz = 2\pi i.$