Weird (?): Use Cauchy's Integral formula to calculate $\int _{|z|=3} \frac{z}{z^2-\pi^2}dz$

Question

Weird (?): Use Cauchy's Integral formula to calculate $\large \int _{|z|=3} \frac{z}{z^2-\pi^2}dz$.

But the function $\frac{z}{z^2-\pi^2}$ is holomorphic on all of $|z|=3$. Am I missing something?

Thanks!