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

# Fourier sine transform

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2082 Chaitra (new course), 4 marks
- Video: [Fourier Sine and Cosine Transform Examples and Solutions By GP Sir](https://www.youtube.com/watch?v=xUKTxstShv0) — Dr.Gajendra Purohit
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/fourier-sine-transform
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the Fourier sine transform of f(x) = e^(−x) and hence use the result in Parseval's identity to show ∫_0^∞ x² dx/(x² + 1)² = π/4.

## Short answer

Its Fourier sine transform is ω/(1+ω²). By Parseval's identity, ∫_0^∞ e^{-2x} dx = (2/π)∫_0^∞ ω²/(1+ω²)² dω = 1/2, yielding ∫_0^∞ x²/(x²+1)² dx = π/4.

## Answer

The Fourier sine transform of f(x) is F_s(ω) = ∫₀^∞ f(x) sin ωx dx, with inverse f(x) = (2/π)∫₀^∞ F_s(ω) sin ωx dω.

**Transform of e^(−x):**

```text
F_s(ω) = ∫₀^∞ e^(−x) sin ωx dx = [ e^(−x)(−sin ωx − ω cos ωx)/(1 + ω²) ]₀^∞
       = ω / (1 + ω²)
```

(In the symmetric convention F_s = √(2/π)·ω/(1+ω²); the result below is the same.)

**Parseval's identity** for sine transforms (with the convention above):

∫₀^∞ [f(x)]² dx = (2/π) ∫₀^∞ [F_s(ω)]² dω

```text
Left side :  ∫₀^∞ e^(−2x) dx = 1/2
Right side:  (2/π) ∫₀^∞ ω² / (1 + ω²)² dω
```

Equating: (2/π) ∫₀^∞ ω²/(1 + ω²)² dω = 1/2, so ∫₀^∞ ω²/(1 + ω²)² dω = π/4.

Renaming the dummy variable ω as x:

**Answer:** ∫₀^∞ x² dx / (x² + 1)² = π/4.

## Check yourself

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

1. F_s of e^(−x) is:
   A. 1/(1 + ω^2)
   B. ω/(1 − ω^2)
   C. ω/(1 + ω^2)
   D. e^(−ω)
   Answer: C. Direct integration. (Key point: Transform)

2. Parseval for sine transforms (this convention):
   A. ∫ f^2 dx = ∫ F_s^2 dω
   B. ∫ f^2 dx = (π/2)∫ F_s^2 dω
   C. ∫ f dx = ∫ F_s dω
   D. ∫ f^2 dx = (2/π)∫ F_s^2 dω
   Answer: D. Factor 2/π. (Key point: Parseval)

3. ∫0^∞ e^(−2x) dx equals:
   A. 1/2
   B. 1
   C. 2
   D. 1/4
   Answer: A. Direct. (Key point: Left side)

4. ∫0^∞ x^2/(x^2+1)^2 dx equals:
   A. π/2
   B. π/4
   C. π/8
   D. 1/2
   Answer: B. (2/π) I = 1/2. (Key point: Result)

## Related in Mathematics III

- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function-two) — Mathematics III, chapter 4
- [Fourier cosine transform](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-cosine-transform) — Mathematics III, chapter 4
- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function) — Mathematics III, chapter 4
- [Half range sine series: π − x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x) — Mathematics III, chapter 4, asked 13×
- [Half range sine series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-three) — Mathematics III, chapter 4, asked 11×
- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-two) — Mathematics III, chapter 4, asked 9×

## Related in other subjects

- [Cosine transform of e^{-x}; Parseval identity](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-parseval-e-x) — Applied Mathematics, chapter 4, asked 3×
- [Sine transform of e^{-x}; ∫x²/(1+x²)²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x2-over-1-x2-sq) — Applied Mathematics, chapter 4, asked 4×
- [Sine transform of e^{-x}; ∫x sin mx/(x²+1)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x-sin-mx) — Applied Mathematics, chapter 4, asked 4×
- [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