---
title: "Solve an integral equation"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:solve-an-integral-equation"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-an-integral-equation
site: BCT Engineering
---

# Solve an integral equation

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/solve-an-integral-equation
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Solve the integral equation ∫_0^∞ f(x) cos αx dx = e^(−α), α > 0.

## Short answer

f(x) = 2/[π(1+x²)]. By Fourier cosine inversion, f(x) = (2/π)∫₀^∞ e^(−α) cos αx dα = 2/(π(1+x²)).

## Answer

Write the equation as a Fourier cosine transform: F_c(α) = ∫₀^∞ f(x) cos αx dx = e^(−α). The inversion formula is f(x) = (2/π)∫₀^∞ F_c(α) cos αx dα.

```text
f(x) = (2/π) ∫₀^∞ e^(−α) cos αx dα
     = (2/π) · [ 1/(1 + x²) ]            using ∫₀^∞ e^(−aα) cos xα dα = a/(a² + x²), a = 1
```

**Answer:** f(x) = 2 / [π(1 + x²)]

Check: ∫₀^∞ 2cos αx/[π(1+x²)] dx = e^(−α), which is the standard cosine transform of 1/(1+x²).

## Check yourself

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

1. The equation is a:
   A. Fourier sine transform
   B. Fourier cosine transform F_c(α) = e^(−α)
   C. Laplace transform
   D. Z-transform
   Answer: B. The kernel is cos αx. (Key point: Identify transform)

2. f(x) is found by:
   A. (2/π)∫ F_c sin αx dα
   B. ∫ F_c dα
   C. (2/π)∫0^∞ F_c(α) cos αx dα
   D. (1/π)∫ F_c dx
   Answer: C. Inverse cosine transform. (Key point: Inversion)

3. f(x) =
   A. 1/(1 + x^2)
   B. 2/[π(1 − x^2)]
   C. e^(−x)
   D. 2/[π(1 + x^2)]
   Answer: D. ∫ e^(−α) cos αx dα = 1/(1 + x^2). (Key point: Result)

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