How to choose a correct contour

Question

In the answer of the question, it select z with radius 1 centred at origin as the contour. But it have two poles and one lies outside from the contour. I am confusing why shouldn't we choose a bigger contour to include the second pole.

Answer 1

Because then after doing the substitution $z=Re^{i\theta}$, we would not obtain the integral that we wish to compute.

Answer 2

Because the contour integral is over $\mid z\mid=1$, which is the unit circle, so the radius $R=1$ and centre is the origin. If $\mid z\mid=a$, then you would have $R=a$. Or if $\mid z-c\mid=a$ you would have $R=a$ and centre $c$.

It could be that later you encounter piecewise contours, e.g. a half circle, or a rectangular contour, or any shape indeed. You would split the integral over these different pieces and parametrize $z$ accordingly as a line segment, a portion of a circle, etc.