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

# Solve an integral equation

Mathematics III · Chapter 4: Fourier Series

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

## Question

Solve the integral equation ∫_0^∞ f(x) sin ωx dx = 1 for 0 ≤ ω < 1; 2 for 1 ≤ ω < 2; 0 for ω ≥ 2, using Fourier transform.

## Short answer

By Fourier sine inversion, f(x) = (2/π) ∫_0^∞ F_s(ω) sin ωx dω = (2/π)[∫_0^1 sin ωx dω + 2∫_1^2 sin ωx dω] = (2/(πx)) (1 + cos x − 2 cos 2x) for x > 0.

## Answer

Write the equation as a Fourier sine transform: F_s(ω) = ∫₀^∞ f(x) sin ωx dx, where F_s(ω) = 1 for 0 ≤ ω < 1, 2 for 1 ≤ ω < 2, and 0 for ω ≥ 2. The inverse sine transform is f(x) = (2/π)∫₀^∞ F_s(ω) sin ωx dω.

```text
f(x) = (2/π) [ ∫₀¹ 1·sin ωx dω + ∫₁² 2·sin ωx dω ]
     = (2/π) [ (1 − cos x)/x + 2(cos x − cos 2x)/x ]
     = (2/πx) [ 1 − cos x + 2cos x − 2cos 2x ]
```

**Answer:** f(x) = (2 / πx) (1 + cos x − 2 cos 2x),  x > 0

Check: as x → 0, 1 + cos x − 2cos 2x → 0, so f stays finite (f(0) = 0 in the limit).

## Check yourself

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

1. The kernel sin ωx means:
   A. cosine transform
   B. Fourier sine transform
   C. Laplace
   D. Mellin
   Answer: B. Standard. (Key point: Identify)

2. f(x) =
   A. (2/π)∫ F_s cos ωx dω
   B. ∫ F_s dω
   C. (2/π)∫0^∞ F_s(ω) sin ωx dω
   D. (1/π)∫ F_s sin ωx dω
   Answer: C. Inverse sine. (Key point: Inverse)

3. The integral splits into:
   A. ∫0^2
   B. 2∫0^1 + ∫1^2
   C. ∫0^∞
   D. ∫0^1 sin ωx dω + 2∫1^2 sin ωx dω
   Answer: D. F_s = 1, 2, 0. (Key point: Splitting)

4. f(x) =
   A. (2/πx)(1 + cos x − 2 cos 2x)
   B. (2/πx)(1 − cos x)
   C. (2/π)(1 + cos x)
   D. (1/πx)(cos x)
   Answer: A. Sum. (Key point: Result)

## Related in Mathematics III

- [Solve an integral equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-an-integral-equation) — Mathematics III, chapter 4
- [Solve ODE by Laplace transform y''+2y'+5y=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-six) — Mathematics III, chapter 3, asked 2×
- [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
- [Solve ODE by Laplace transform x''+2x'+5x=e^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-x) — Mathematics III, chapter 3, asked 17×
- [Solve ODE by Laplace transform y''+2y'+2y=5sinx](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-17) — Mathematics III, chapter 3, asked 17×

## Related in other subjects

- [Fourier transform of 1-x²; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-1-x2-evaluate) — Applied Mathematics, chapter 4, asked 2×
- [Solve sin x dy/dx + y cos x = x sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-sinx-dy-dx-y-cosx) — Mathematics I, chapter 4
- [DUIS: ∫ e^(−x) sin bx / x dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-e-x-sin-bx-over-x) — Mathematics I, chapter 2, asked 10×
- [DUIS: ∫ log((a+b sin x)/(a−b sin x)) dx/sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-log-a-plus-b-sinx) — Mathematics I, chapter 2, asked 9×