---
title: "Sine transform of e^{-x}; ∫x²/(1+x²)²"
subject: "Applied Mathematics"
chapter: 4
key: "semester_4/applied_mathematics:fourier-sine-transform-x2-over-1-x2-sq"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-sine-transform-x2-over-1-x2-sq
site: BCT Engineering
---

# Sine transform of e^{-x}; ∫x²/(1+x²)²

Applied Mathematics · Chapter 4: Fourier Transform

- Asked in: 2073 Magh, 5 marks; 2068 Bhadra (asked 2 times)
- Video: [9.4 Parseval's Identity for Fourier Transform](https://www.youtube.com/watch?v=7044htChxec) — Laxman Verma
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/fourier-sine-transform-x2-over-1-x2-sq
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-4

## Question

Find the Fourier sine transform of e^{-x}, x ≥ 0 and hence show that (evaluate) ∫₀^∞ x²/(1 + x²)² dx = π/4.

## Short answer

F_s(s)=√(2/π) s/(1+s²). By Parseval’s identity, ∫₀^∞ e^{-2x} dx = 1/2 equals (2/π)∫₀^∞ s²/(1+s²)² ds. Hence ∫₀^∞ s²/(1+s²)² ds = π/4. Rename s→x: ∫₀^∞ x²/(1+x²)² dx = π/4.

## Answer

**Definition.** F_s(s) = √(2/π) ∫₀^∞ f(x) sin sx dx.

#### Sine transform of e⁻ˣ

Using ∫₀^∞ e^{−ax} sin sx dx = s/(a² + s²) with a = 1:

**F_s(s) = √(2/π) · s/(1 + s²)**

#### Using Parseval's identity

For sine transforms, ∫₀^∞ [f(x)]² dx = ∫₀^∞ [F_s(s)]² ds.

- Left side: ∫₀^∞ e^{−2x} dx = ½
- Right side: (2/π) ∫₀^∞ s²/(1 + s²)² ds

```text
(2/π) ∫₀^∞ s²/(1 + s²)² ds = 1/2
∫₀^∞ s²/(1 + s²)² ds       = π/4
```

Renaming the variable s to x:

**∫₀^∞ x²/(1 + x²)² dx = π/4** ∎

(Numerical check: 0.785398 ✓. This also agrees with ∫₀^∞ dx/(1 + x²) − ∫₀^∞ dx/(1 + x²)² = π/2 − π/4.)

## Check yourself

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

1. What is Fs of e^{-x}?
   A. sqrt(2/pi) 1/(1 + s^2)
   B. s/(1+s^2)
   C. sqrt(2/pi) s/(1 - s^2)
   D. sqrt(2/pi) s/(1 + s^2)
   Answer: D. int e^{-x} sin sx dx = s/(1+s^2). (Key point: Sine transform)

2. Which identity gives the integral?
   A. Parseval: int f^2 dx = int Fs^2 ds
   B. Inversion formula
   C. Convolution theorem
   D. Fourier integral at x=0
   Answer: A. Square of Fs gives s^2/(1+s^2)^2. (Key point: Parseval)

3. What equation follows?
   A. (2/pi) int s/(1+s^2) ds = 1
   B. (2/pi) int s^2/(1+s^2)^2 ds = 1/2
   C. (1/pi) int s^2/(1+s^2)^2 ds = 1
   D. int s^2/(1+s^2)^2 ds = 1/2
   Answer: B. The left side int e^{-2x} dx = 1/2. (Key point: Equation)

## Close variants set in other exams

- [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×

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