Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 2 / 4 / i / Solution

For a counterclockwise simple contour containing $z_0$, and $f$ analytic on a neighbourhood of its interior and boundary, the Cauchy integral formula isFor the required rational integral, use partial fraction decomposition:Both points $1,-1$ lie inside $|z|=2$. Applying the formula with the constant analytic function one gives $2\pi i$ for each of the two terms. They cancel, soReversing the circle's orientation would leave this zero result unchanged.