Final answer:
To compute the integral, we need to parameterize the given curve and then simplify the integral. By factoring out 9 from the denominator and using the partial fraction decomposition, we can split the integrand into two separate fractions.
Step-by-step explanation:
To compute the integral ∫2/z² dz along the given curve, we need to parameterize the curve. Let z = -3 + 3i*t, where t goes from 0 to 1. Then dz = 3i*dt, and substituting these values into the integral, we get:
∫2/z² dz = ∫2/(-3 + 3i*t)² * 3i*dt = ∫2/(9 - 18i*t - 9t²) * 3i*dt
Next, we can factor out 9 from the denominator and simplify the integral:
∫2/(9(1 - 2i*t - t²)) * 3i*dt = ∫2/(9(1 - ( √2*i*t - t²))) * 3i*dt
Using the partial fraction decomposition, we can split the integrand into two separate fractions and integrate each term separately:
∫2/(9(1 - ( √2*i*t - t²))) * 3i*dt = ∫2/(9(1 - √2*i*t + t²)) * 3i*dt = ∫2/((3 - √2*i*t + 3i*t)(3 + √2*i*t + 3i*t)) * 3i*dt