---
title: "Evaluate a line integral"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-a-line-integral-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-two
site: BCT Engineering
---

# Evaluate a line integral

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Chaitra (new course), 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-a-line-integral-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

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

## 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.

## Answer

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:

```text
∮_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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The integrand's singularity is:
   A. z = 1, inside |z| = 2
   B. z = 2 on the circle
   C. z = -1 inside
   D. none
   Answer: A. Denominator vanishes at z=1 and |1|<2. (Key point: Pole)

2. The formula used is:
   A. integral = g(a)
   B. Cauchy's integral formula: integral g/(z-a) = 2 pi i g(a)
   C. integral = pi i g(a)
   D. integral = 0
   Answer: B. g(z) = 5z - 2 is analytic inside. (Key point: Formula)

3. The value is:
   A. 3 pi i
   B. 2 pi i
   C. 6 pi i
   D. 0
   Answer: C. 2 pi i * g(1) = 2 pi i * 3. (Key point: Value)

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