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

# Fourier integral representation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 2 marks
- Video: [Fourier Integral | Fourier Cosine and Sine Integral | Important Problems | Engineering Mathematics](https://www.youtube.com/watch?v=CEgIgY4z66E) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-integral-representation-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier integral representation of f(x) = e^(−|x|).

## Short answer

Since e^{-|x|} is even, its Fourier integral is (2/π)∫_0^∞ cos(αx)/(1+α²) dα, giving the representation e^{-|x|} = (2/π)∫_0^∞ cos(αx)/(1+α²) dα.

## Answer

The function e^(−|x|) is even, so its Fourier integral is f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos αv dv] cos αx dα.

```text
∫₀^∞ e^(−v) cos αv dv = 1/(1 + α²)
```

**Fourier integral representation**

e^(−|x|) = (2/π) ∫₀^∞ cos αx / (1 + α²) dα,  −∞ < x < ∞

The function is continuous everywhere, so the integral equals e^(−|x|) at every x. At x = 0 it gives ∫₀^∞ dα/(1 + α²) = π/2.

## Check yourself

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

1. e^(−|x|) is even, so the integral uses:
   A. sine
   B. both equally
   C. cosine
   D. exponentials only
   Answer: C. Even function. (Key point: Symmetry)

2. ∫0^∞ e^(−v) cos αv dv =
   A. α/(1 + α^2)
   B. 1/(1 − α^2)
   C. e^(−α)
   D. 1/(1 + α^2)
   Answer: D. Standard. (Key point: Inner)

3. e^(−|x|) =
   A. (2/π)∫0^∞ cos αx/(1 + α^2) dα
   B. (1/π)∫ sin αx/(1 + α^2) dα
   C. (2/π)∫ α sin αx/(1+α^2) dα
   D. (2/π)∫ cos αx dα
   Answer: A. Substitute. (Key point: Result)

## Close variants set in other exams

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

## Related in Mathematics III

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

## Related in other subjects

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- [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×
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