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Evaluate a line integral

Short answer

The integrand has a simple pole at z=1 inside |z|=2. By Cauchy's integral formula with g(z)=5z-2, the integral equals 2πi·g(1)=2πi·3=6πi.

Evaluate ∮_C (5z − 2)/(z − 1) dz, C being the circle |z| = 2.

Let f(z) = (5z − 2)/(z − 1). It has a simple pole at z = 1, which lies inside the circle |z| = 2.

By Cauchy's integral formula, with g(z) = 5z − 2 analytic inside C:

∮_C g(z)/(z − 1) dz = 2πi · g(1) = 2πi (5 − 2) = 6πi

(Equivalently, the residue at z = 1 is 3 and the integral is 2πi × 3.)

Answer: ∮_C (5z − 2)/(z − 1) dz = 6πi.

✓ Answer checked by Claude Opus 5.5 (high effort)

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