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

# Fourier integral representation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Kartik (new course), 4 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Obtain the Fourier integral of the function f(x) = e^(−x) for x > 0; 0 for x < 0. Hence evaluate ∫_0^∞ 1/(1 + x²) dx.

## Short answer

Fourier integral representation: f(x) = (1/π) ∫_0^∞ (cos αx + α sin αx)/(1+α^2) dα. Setting x = 0 where f(0) averages 1/2 gives 1/2 = (1/π)∫_0^∞ dα/(1+α^2), so ∫_0^∞ dx/(1+x^2) = π/2.

## Answer

The Fourier integral of f(x) is f(x) = ∫₀^∞ [A(α) cos αx + B(α) sin αx] dα, with A(α) = (1/π)∫ f(v) cos αv dv and B(α) = (1/π)∫ f(v) sin αv dv.

**Coefficients.** f(v) = e^(−v) for v > 0, and 0 for v < 0.

```text
A(α) = (1/π)∫₀^∞ e^(−v) cos αv dv = 1 / [π(1 + α²)]
B(α) = (1/π)∫₀^∞ e^(−v) sin αv dv = α / [π(1 + α²)]
```

**Fourier integral**

f(x) = (1/π) ∫₀^∞ (cos αx + α sin αx) / (1 + α²) dα

This equals e^(−x) for x > 0, 0 for x < 0, and the mean value 1/2 at the jump x = 0.

**Evaluation of ∫₀^∞ dx/(1 + x²).** Put x = 0 in the integral. The value there is the average (0 + 1)/2 = 1/2:

1/2 = (1/π) ∫₀^∞ dα / (1 + α²)

**Answer:** ∫₀^∞ dx / (1 + x²) = π/2.

## Check yourself

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

1. A(α) = (1/π)∫ f(v) cos αv dv and B(α) = (1/π)∫ f(v) sin αv dv, for f(v) = e^(−v), v>0 give:
   A. A = α/..., B = 1/...
   B. A = B
   C. A = 1/[π(1+α^2)], B = α/[π(1+α^2)]
   D. A = 0
   Answer: C. Standard integrals. (Key point: Formula)

2. At x = 0 the integral equals:
   A. 1
   B. 0
   C. e^0 = 1
   D. 1/2
   Answer: D. Mean of 0 and 1. (Key point: Jump)

3. ∫0^∞ dx/(1+x^2) =
   A. π/2
   B. π
   C. π/4
   D. 1
   Answer: A. 1/2 = (1/π)∫. (Key point: Result)

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