Let C denote the circle |z| = 1, taken counterclockwise, and use the following steps to show that
int(exp(z + 1/z) dz) = i2pie sum(1/ (n!(n + 1)!)
1) By using maclaurin series for e^z write the above integral as
sum(1/n! int(z^n exp(1/z) dz) )
2) Apply cauchy's residue theorem to evaluate the integrals above.