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

# Fourier cosine integral

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2081 Chaitra (new course), 2 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier cosine integral of f(x) = e^(−ax) for x > 0, a > 0.

## Short answer

The Fourier cosine integral of e^(-ax) for x>0 is e^(-ax) = (2/π) ∫_0^∞ [a/(a²+ω²)] cos ωx dω. Equivalently, ∫_0^∞ cos ωx/(a²+ω²) dω = π e^(-ax)/(2a).

## Answer

For x > 0 the Fourier cosine integral is f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos ωv dv] cos ωx dω.

With f(v) = e^(−av): ∫₀^∞ e^(−av) cos ωv dv = a/(a² + ω²).

**Answer:**

e^(−ax) = (2/π) ∫₀^∞ [a / (a² + ω²)] cos ωx dω,  x > 0, a > 0

(Equivalently ∫₀^∞ cos ωx/(a² + ω²) dω = π e^(−ax)/(2a).)

## Check yourself

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

1. Cosine integral of f(x), x>0:
   A. (2/π)∫ [∫ f(v) cos ωv dv] cos ωx dω
   B. (1/π)∫ ... sin
   C. ∫ f dv
   D. (2/π)∫ f sin
   Answer: A. Standard. (Key point: Formula)

2. ∫0^∞ e^(−av) cos ωv dv =
   A. ω/(a^2+ω^2)
   B. a/(a^2 + ω^2)
   C. 1/(a^2+ω^2)
   D. a/(a+ω)
   Answer: B. Standard. (Key point: Inner)

3. ∫0^∞ cos ωx/(a^2+ω^2) dω =
   A. π e^(−ax)/a^2
   B. 2e^(−ax)/(πa)
   C. π e^(−ax)/(2a)
   D. e^(−ax)
   Answer: C. Rearrange. (Key point: Result)

## Related in Mathematics III

- [Fourier cosine integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-integral) — Mathematics III, chapter 4
- [Fourier cosine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-transform) — Mathematics III, chapter 4
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- [Fourier integral representation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-integral-representation-four) — Mathematics III, chapter 4

## Related in other subjects

- [Fourier cosine integral of e^{-kx}](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-integral-e-kx) — Applied Mathematics, chapter 4, asked 2×
- [Cosine transform of e^{-mx}; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-transform-e-mx-beta) — Applied Mathematics, chapter 4, asked 3×
- [Fourier integral of e^{-x}; ∫cos sx/(s²+1)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-integral-e-x-cos-sx) — Applied Mathematics, chapter 4, asked 3×
- [Fourier sine integral of e^{-x}cos x](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-integral-e-x-cosx) — Applied Mathematics, chapter 4