Evaluate the integral around C of [(e^3z)/(z+pi/2)] dz if C is the circle |z| = 5.
$\displaystyle f(z)$ has a simple pole at $\displaystyle -\frac{pi}{2}$
so the residue is $\displaystyle \lim_{z}^{-\frac{\pi}{2}}\left( z+\frac{\pi}{2}\right)f(z)=e^{-\frac{3\pi}{2}}$
so
$\displaystyle \oint_{|z|=5} \frac{e^{3z}}{z+\frac{\pi}{2}}dz=2\pi i e^{-\frac{3\pi}{2}}=2\pi i (\cos\left(\frac{3\pi}{2} \right)+i\sin\left(\frac{3\pi}{2} \right))=2\pi$