---
title: "Contour ∫dθ/(cosθ+2)"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:contour-1-over-cos-plus-2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/contour-1-over-cos-plus-2
site: BCT Engineering
---

# Contour ∫dθ/(cosθ+2)

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2071 Magh, 5 marks; 2070 Magh, 5 marks; 2068 Bhadra (asked 3 times)
- Video: [By Using Contour Integration Type 01 Problems 02 -Complex Integration](https://www.youtube.com/watch?v=dvwFlCOqrmA) — SK Maths Tutorial
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/contour-1-over-cos-plus-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Evaluate ∫₀^{2π} 1/(cosθ + 2) dθ by contour integration in the complex plane.

## Short answer

Put z=e^{iθ}; the integral becomes (2/i)∮ dz/(z²+4z+1) around |z|=1. Only pole inside is −2+√3 with residue 1/(2√3); by the residue theorem I=2π/√3.

## Answer

**Method.** For an integral over 0 to 2π of a function of cos θ, put z = e^{iθ}. Then the interval becomes the unit circle |z| = 1 and

cos θ = (z + 1/z)/2,  dθ = dz/(iz).

#### Solution
cos θ + 2 = (z + 1/z)/2 + 2 = (z² + 4z + 1)/(2z)

```text
I = ∮_C [2z/(z² + 4z + 1)] · dz/(iz)
  = (2/i) ∮_C dz/(z² + 4z + 1),   C: |z| = 1
```

Poles: z² + 4z + 1 = 0 gives z = −2 ± √3.

- z = α = −2 + √3 ≈ −0.268: inside C (|α| < 1)
- z = β = −2 − √3 ≈ −3.732: outside C

Residue at z = α (simple pole):

Res = 1/(α − β) = 1/(2√3)

By the residue theorem:

I = (2/i) · 2πi · 1/(2√3) = 2π/√3

**Answer: ∫₀^{2π} dθ/(cos θ + 2) = 2π/√3 ≈ 3.628**

(Check with the standard result ∫₀^{2π} dθ/(a + cos θ) = 2π/√(a² − 1) for a > 1; here a = 2.)

## Check yourself

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

1. For ∫₀^{2π} F(cos θ) dθ we put z = e^{iθ}, giving dθ =
   A. dz/(iz)
   B. i dz/z
   C. dz/z
   D. iz dz
   Answer: A. dz = iz dθ. (Key point: Substitution)

2. cos θ + 2 in terms of z becomes:
   A. (z² + 4z + 1)/z
   B. (z² − 4z + 1)/(2z)
   C. (z² + 2z + 1)/(2z)
   D. (z² + 4z + 1)/(2z)
   Answer: D. (z + 1/z)/2 + 2 = (z²+4z+1)/(2z). (Key point: Transformed integral)

3. Which pole of z² + 4z + 1 = 0 lies inside |z| = 1?
   A. 2 + √3
   B. −2 + √3
   C. −1
   D. −2 − √3
   Answer: B. −2+√3 ≈ −0.268 is inside; −3.732 is outside. (Key point: Pole inside)

4. The integral equals:
   A. 4π/√3
   B. π/√3
   C. 2π√3
   D. 2π/√3
   Answer: D. I = (2/i)(2πi)/(2√3) = 2π/√3. (Key point: Result)

## Close variants set in other exams

- [Contour ∫dθ/(√2-cosθ)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acontour-1-over-sqrt2-cos) — Applied Mathematics, chapter 1, asked 4×

## Related in Applied Mathematics

- [Define complex integral; evaluate ∮log z dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-log-z) — Applied Mathematics, chapter 1
- [Contour integral ∫x²/(1+x²)³](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Areal-integral-x2-over-1-x2-cubed) — Applied Mathematics, chapter 1, asked 4×
- [Contour ∫cos3θ/(5-4cosθ) dθ](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acontour-cos3theta-over-5-4cos) — Applied Mathematics, chapter 1, asked 3×
- [Contour ∫dθ/(5-4sinθ)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acontour-1-over-5-4sin) — Applied Mathematics, chapter 1
- [Contour integral ∫x²/((x²+a²)(x²+b²))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Areal-integral-x2-a-b) — Applied Mathematics, chapter 1, asked 3×
- [Contour integral ∫x²/((1+x²)(x²+4))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Areal-integral-x2-1-x2-x2-4) — Applied Mathematics, chapter 1, asked 2×

## Related in other subjects

- [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×
- [Real integral by contour integration](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Areal-integral-by-contour-integration-two) — Mathematics III, chapter 2, asked 2×
- [Change order of integration: cos y/√((a−x)(a−y))](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Achange-order-integration-cos-y) — Mathematics, chapter 2
- [Gauss quadrature numerical integration](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Agauss-quadrature-integration) — Finite Element Method, chapter 7