---
title: "Romberg: 1/x² from 1 to 2"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:romberg-1-over-x2-1-2"
source: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-1-over-x2-1-2
site: BCT Engineering
---

# Romberg: 1/x² from 1 to 2

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2068 Magh, 6 marks
- Video: [Romberg's Method l Numerical Methods](https://www.youtube.com/watch?v=ctfkw-oq8kY) — SK Maths Tutorial
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-1-over-x2-1-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

Use Romberg's method to compute ∫ from 1 to 2 of dx/(1 + x²).

## Short answer

Applying Richardson extrapolation to trapezoidal values with h=1, 0.5, 0.25, 0.125 yields the Romberg estimate 0.3218.

## Answer

Romberg's method applies Richardson extrapolation, R(i,j) = [4ʲ·R(i,j−1) − R(i−1,j−1)] / (4ʲ − 1), to the trapezoidal values T(h) with h = 1, 0.5, 0.25, 0.125.

f(x) = 1/(1+x²), a = 1, b = 2.

#### Trapezoidal values
```text
T(1)     = (1/2)[f(1)+f(2)]                          = 0.3500
T(0.5)   = ½T(1) + 0.5·f(1.5)                        = 0.3288
T(0.25)  = ½T(0.5) + 0.25[f(1.25)+f(1.75)]           = 0.3235
T(0.125) = ½T(0.25) + 0.125[f(1.125)+f(1.375)+f(1.625)+f(1.875)] = 0.3222
```

#### Romberg table
| h | T (order h²) | R1 (order h^4) | R2 (order h^6) | R3 (order h^8) |
|---|---|---|---|---|
| 1.0000 | 0.3500 |  |  |  |
| 0.5000 | 0.3288 | 0.3218 |  |  |
| 0.2500 | 0.3235 | 0.3217 | 0.3217 |  |
| 0.1250 | 0.3222 | 0.3218 | 0.3218 | 0.3218 |

**Answer: ∫₁² dx/(1+x²) ≈ 0.3218** (exact value tan⁻¹2 − tan⁻¹1 = 0.3218).

## Check yourself

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

1. Which integrand is used?
   A. 1/x²
   B. 1/(1-x²)
   C. x²/(1+x²)
   D. 1/(1+x²) as given in the question
   Answer: D. The title says 1/x² but the stated function is used. (Key point: Function)

2. How is T(1) found on [1, 2]?
   A. (1/2)[f(1)+f(2)] = 0.3500
   B. f(1.5)
   C. f(1)+f(2)
   D. 0.3218
   Answer: A. f(1) = 0.5, f(2) = 0.2. (Key point: T(1))

3. What does Richardson extrapolation do?
   A. Averages them
   B. Combines two trapezoid estimates to cancel the leading h² error
   C. Takes the larger
   D. Doubles h
   Answer: B. R1 = (4T(h/2)-T(h))/3. (Key point: Extrapolation)

4. What is the result?
   A. 0.3500
   B. 0.3222
   C. 0.3218
   D. 0.5
   Answer: C. Converged R3 = 0.3218 = atan2 - atan1. (Key point: Result)

## Related in Numerical Method

- [Romberg: 1/(1+x²) from -1 to 1](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aromberg-1-over-1-x2) — Numerical Method, chapter 5
- [Romberg: (e^x + sin x)/(1+x²)](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aromberg-exp-sin-over-1-x2) — Numerical Method, chapter 5, asked 2×
- [Romberg: e^(-x²) from 1 to 2](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aromberg-e-x2-1-2) — Numerical Method, chapter 5