---
title: "Real integral by contour integration"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:real-integral-by-contour-integration-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/real-integral-by-contour-integration-two
site: BCT Engineering
---

# Real integral by contour integration

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

- Asked in: 2081 Chaitra (new course), 4 marks
- Video: [Contour Integration.  Type :1 ( Trigonometric Functions)  in Complex Analysis.](https://www.youtube.com/watch?v=4Ze5-9jUCvU) — Maths preparation corner
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/real-integral-by-contour-integration-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∫_0^{2π} dθ/(5 + 4 sin θ) using the method of contour integration in the complex plane.

## Short answer

Substitute z = e^(iθ), sinθ = (z−1/z)/(2i), dθ = dz/(iz). The contour integral over unit circle has a pole at z = −i/2 inside, residue = 1/(3i). By residue theorem, I = 2π/3.

## Answer

Let I = ∫₀^2π dθ/(5 + 4 sin θ). Put z = e^(iθ) on |z| = 1: sin θ = (z − 1/z)/(2i), dθ = dz/(iz).

```text
5 + 4 sin θ = 5 + (2/i)(z − 1/z) = 5 − 2iz + 2i/z
            = (−2i z² + 5z + 2i)/z
I = ∮_C [z/(−2i z² + 5z + 2i)] · dz/(iz)
  = ∮_C dz / [i(−2i z² + 5z + 2i)]
  = ∮_C dz / (2z² + 5iz − 2)
```

Factor: 2z² + 5iz − 2 = 2(z + 2i)(z + i/2). Poles: z = −2i (outside the unit circle) and z = −i/2 (inside).

```text
Res at z = −i/2 = 1 / [2(z + 2i)] at z = −i/2 = 1/(2 · 3i/2) = 1/(3i)
I = 2πi × 1/(3i) = 2π/3
```

**Answer: I = 2π/3.** (Check: the standard formula 2π/√(a² − b²) = 2π/√(25 − 16) = 2π/3.)

## Check yourself

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

1. On |z|=1, sin theta =
   A. (z + 1/z)/2
   B. (z - 1/z)/2
   C. (z - 1/z)/(2i)
   D. (z + 1/z)/(2i)
   Answer: C. Euler. (Key point: Substitution)

2. The integral becomes:
   A. contour of dz/(z^2+5)
   B. contour of z dz
   C. contour of dz/(5+4z)
   D. contour of dz/(2z^2 + 5iz - 2)
   Answer: D. Algebra shown. (Key point: Integrand)

3. Pole inside unit circle:
   A. z = -i/2
   B. z = -2i
   C. z = 2i
   D. z = i/2
   Answer: A. |-2i|=2>1. (Key point: Pole inside)

4. The value is:
   A. pi/3
   B. 2 pi/3
   C. 2 pi/9
   D. 4 pi/3
   Answer: B. 2 pi i * 1/(3i); matches 2pi/sqrt(25-16). (Key point: Result)

## Close variants set in other exams

- [Real integral by contour integration](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Areal-integral-by-contour-integration) — Mathematics III, chapter 2, asked 2×

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