Final answer:
The question involves evaluating a complex integral using the residue theorem, by finding and summing the residues of the function within the circle of radius e centered at -2i.
Step-by-step explanation:
The student is asked to evaluate a complex integral over a closed path C, which is a circle of radius e centered at -2i. To solve this integral, we must consider residues within the circle C as described by the residue theorem from complex analysis.
The given function can be rewritten and simplified to find its poles and corresponding residues. One should identify the poles that lie inside the circle C and then calculate the residues at those poles. After that, the integral can be evaluated using the sum of these residues multiplied by 2πi according to the residue theorem.