How to use Cauchy integral to evaluate the integral $\int_C \frac{\cos\ z}{z(z^2+8)}dz$. The integral in question is: > $$\large{\int_C \frac{\cos\ z}{z(z^2+8)}dz}$$ Where $C$ is the square whose vertices are $x=\pm2...--prophetes.ai

How to use Cauchy integral to evaluate the integral $\int_C \frac{\cos\ z}{z(z^2+8)}dz$. The integral in question is: > $$\large{\int_C \frac{\cos\ z}{z(z^2+8)}dz}$$ Where $C$ is the square whose vertices are $x=\pm2, y=\pm2$, anti-clockwise direction. I did the natural thing to split them up using partial fraction decomposition, but I am stuck immediately. Any help/insights is deeply appreciated.

**HINT:**

$$\frac{1}{z(z^2+8)}=\frac{1/8}{z}-\frac{z/8}{z^2+8}=\frac{1/8}{z}-\frac{1/16}{z+i2\sqrt 2}-\frac{1/16}{z-i2\sqrt 2}$$

Note that the only pole enclosed by the square contour is the one at $z=0$.