---
title: "Fourier cosine integral"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:fourier-cosine-integral"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-cosine-integral
site: BCT Engineering
---

# Fourier cosine integral

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 4 marks
- Video: [How to solve Fourier cosine integral? || How to solve Fourier sine integral?|| Fourier integral ||](https://www.youtube.com/watch?v=5i6Ptd4t_jg) — Mathematics Solutions Dr Mohan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-cosine-integral
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier cosine integral of f(x) = e^(−kx), k > 0, x > 0 and use it to evaluate ∫_0^∞ cos ωx/(k² + ω²) dω.

## Short answer

Using Fourier cosine integral, e^(-kx) = (2/π) ∫_0^∞ [k/(k²+ω²)] cos ωx dω. Hence ∫_0^∞ cos ωx/(k²+ω²) dω = π/(2k) e^(-kx), for x ≥ 0.

## Answer

The Fourier cosine integral of an even extension of f(x), x > 0, is f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos ωv dv] cos ωx dω.

**Inner integral** for f(v) = e^(−kv):

```text
∫₀^∞ e^(−kv) cos ωv dv = k/(k² + ω²)
```

**Fourier cosine integral**

e^(−kx) = (2/π) ∫₀^∞ [k/(k² + ω²)] cos ωx dω,  x > 0, k > 0

**Evaluation.** Rearranging the above,

∫₀^∞ cos ωx / (k² + ω²) dω = (π/2k) e^(−kx),  x ≥ 0

In particular, for x = 0 the integral ∫₀^∞ dω/(k² + ω²) = π/(2k).

## Check yourself

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

1. The cosine integral is:
   A. (1/π)∫ ... sin ωx dω
   B. (2/π)∫ f dv
   C. (2/π)∫0^∞ [∫0^∞ f(v) cos ωv dv] cos ωx dω
   D. ∫ f cos ωx dx
   Answer: C. Standard. (Key point: Formula)

2. ∫0^∞ e^(−kv) cos ωv dv equals:
   A. ω/(k^2 + ω^2)
   B. 1/(k^2 + ω^2)
   C. k/(k + ω)
   D. k/(k^2 + ω^2)
   Answer: D. Standard Laplace form. (Key point: Inner integral)

3. ∫0^∞ cos ωx/(k^2 + ω^2) dω equals:
   A. (π/2k) e^(−kx)
   B. (π/2) e^(−kx)
   C. π e^(−kx)/k^2
   D. (2/πk) e^(−kx)
   Answer: A. Rearrange the integral. (Key point: Evaluation)

4. At x = 0 the integral is:
   A. π/k
   B. π/(2k)
   C. π/2
   D. 0
   Answer: B. e^0 = 1. (Key point: x = 0)

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