Cauchy’s integral theorem is applicable for evaluating integrals over closed contours when the function is analytic inside the contour, while the residue theorem is used for functions with singularities, allowing for the calculation of integrals around poles. The residue theorem is particularly advantageous when dealing with multiple poles, as it simplifies calculations significantly. For the integral of 1/z, both theorems can be applied, but the residue theorem is often preferred for its ease of computation. Understanding the conditions for each theorem is crucial; Cauchy’s requires analyticity, whereas the residue theorem requires identification of poles and their residues. Mastery of these concepts enhances proficiency in complex analysis.