# How to apply the Cauchy Integral Formula to a closed contour integral

$$\oint_{|z|=3}\frac{e^{iz}}{(z^2+1)^2} \mathrm d z$$ The question requires that this integral be solved using Cauchy's Integral Formula, but there seems to be two singularities within the contour at $z=i$ and $z=-i$.

Kindly please tell me how to apply the formula in this case.

• Welcome to MSE. Please type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Apr 25 '18 at 8:27

You can use the fact that$$\frac1{(z^2+1)^2}=\frac1{(z+i)^2(z-i)^2}=\frac14\left(\frac{-2+zi}{(z+i)^2}-\frac{2+zi}{(z-i)^2}\right),$$in order to deal with two distinct singularities.