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

# Fourier integral representation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier integral representation of f(x) = 1 for |x| < 1; 0 for |x| > 1.

## Short answer

f(x) = (2/π) ∫₀^∞ (sin α cos αx)/α dα. The even function's Fourier integral reduces to cosine form with inner integral sin α/α over [0,1].

## Answer

For f(x) defined on (−∞, ∞), the Fourier integral is f(x) = (1/π)∫₀^∞ ∫₋∞^∞ f(v) cos α(x − v) dv dα. Because the given f is even, this reduces to f(x) = (2/π)∫₀^∞ [∫₀^∞ f(v) cos αv dv] cos αx dα.

**Inner integral.** f(v) = 1 for |v| < 1 and 0 otherwise:

```text
∫₀^∞ f(v) cos αv dv = ∫₀¹ cos αv dv = sin α / α
```

**Representation**

f(x) = (2/π) ∫₀^∞ (sin α cos αx / α) dα

This equals 1 for |x| < 1, 0 for |x| > 1, and 1/2 at x = ±1 (the average at the jump).

Putting x = 0 gives the useful result ∫₀^∞ (sin α / α) dα = π/2.

## Check yourself

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

1. For an even f the Fourier integral reduces to:
   A. (2/π)∫ [∫ f sin αv dv] sin αx dα
   B. (1/π)∫ [∫ f dv] dα
   C. (2/π)∫0^∞ [∫0^∞ f(v) cos αv dv] cos αx dα
   D. (2/π)∫ f(v) dv
   Answer: C. Even functions give the cosine form. (Key point: Even reduction)

2. ∫0^1 cos αv dv equals:
   A. cos α/α
   B. α sin α
   C. 1/α
   D. sin α/α
   Answer: D. Direct integration. (Key point: Inner integral)

3. At x = ±1 the integral equals:
   A. 1/2
   B. 1
   C. 0
   D. 2
   Answer: A. Average at the jump. (Key point: Jump value)

## Close variants set in other exams

- [Fourier integral representation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-integral-representation-two) — 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

- [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 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×
- [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
- [Fourier integral of rectangular pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-integral-rectangular-pulse) — Applied Mathematics, chapter 4