Question

Evaluate \( \int_C \frac{dz}{z^2 + 9} \), where \( C \) is \( |z - 3| = 4 \)

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)

Hint:

For integrals involving rational functions, the residue theorem is often used to evaluate integrals over closed contours.

Answer 1

The Correct Option is C

The integral is evaluated using the residue theorem. First, we express the function \( \frac{1}{z^2 + 9} \) as \( \frac{1}{(z - 3i)(z + 3i)} \). The residue at \( z = 3i \) is \( \frac{1}{6i} \), and thus the integral is \( 2\pi i \times \frac{1}{6i} = \frac{\pi}{3} \).