Final Answer:
(a) \(0\)
(b) \(2\pi i\)
(c) \(8\pi i\)
Step-by-step explanation:
For part (a), using Cauchy's Integral Theorem, which states that for a function analytic in a simply connected domain and its closed contour, the integral along the closed contour is \(0\), we note that \(f(z) = z^{-5}/z^2\) is analytic inside and on the contour \(C: |z| = 2\) (excluding \(z = 0\)). Thus, by Cauchy's Integral Theorem, the integral evaluates to \(0\).
In part (b), using Cauchy's Residue Theorem, which states that for a function analytic except for isolated singularities within a closed contour, the integral around the closed contour equals \(2\pi i\) times the sum of the residues inside the contour, we identify the singularity of \(f(z) = e^z/(2z+1)\) at \(z = -\frac{1}{2}\) inside the contour \(C: |z| = 3\). Calculating the residue at \(z = -\frac{1}{2}\) and applying the residue theorem yields the result of \(2\pi i\).
For part (c), applying Cauchy's Integral Theorem to Cauchy's Integral Formula, considering \(f(z) = z^3 + 4z^2/\cos z\), which is analytic inside and on the contour \(C: |z - 3i| = 2\) (excluding poles within), and noting that \(z = 3i\) is outside the contour, the integral evaluates to \(8\pi i\) by Cauchy's Integral Theorem.
In summary, by employing Cauchy's Integral Theorem and Cauchy's Residue Theorem, the line integrals in (a), (b), and (c) are evaluated accordingly to \(0\), \(2\pi i\), and \(8\pi i\) respectively, due to the properties and analytical nature of the functions and contours involved.