---
title: "Romberg: (e^x + sin x)/(1+x²)"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:romberg-exp-sin-over-1-x2"
source: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-exp-sin-over-1-x2
site: BCT Engineering
---

# Romberg: (e^x + sin x)/(1+x²)

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2074 Bhadra, 8 marks; 2070 Bhadra, 6 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/romberg-exp-sin-over-1-x2
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

Evaluate the following integral using Romberg's method (correct to two decimal places): ∫ from 0 to 2 of (e^x + sin x)/(1 + x²) dx

## Short answer

Using Romberg integration with h=2,1,0.5,0.25, trapezoidal estimates are extrapolated. The final converged value to two decimal places is ∫₀² (e^x+ sin x)/(1+x²) dx ≈ 3.29.

## Answer

Romberg integration improves the trapezoidal estimates T(h) by Richardson extrapolation:

R(k, j) = [4ʲ·R(k+1, j−1) − R(k, j−1)] / (4ʲ − 1)

Here f(x) = (eˣ + sin x)/(1 + x²) on [0, 2], with h = 2, 1, 0.5, 0.25.

#### Function values (h = 0.25)

| x | f(x) |
|---|---|
| 0 | 1.0000 |
| 0.25 | 1.4413 |
| 0.5 | 1.7025 |
| 0.75 | 1.7911 |
| 1 | 1.7799 |
| 1.25 | 1.7324 |
| 1.5 | 1.6859 |
| 1.75 | 1.6587 |
| 2 | 1.6597 |

#### Trapezoidal estimates T = h[f₀/2 + f₁ + … + fₙ₋₁ + fₙ/2]

```text
h = 2   : T = 2[ f(0)/2 + f(2)/2 ]                       = 2.6597
h = 1   : T = 1[ f(0)/2 + f(1) + f(2)/2 ]                 = 3.1097
h = 0.5 : T                                              = 3.2491
h = 0.25: T                                              = 3.2804
```

#### Romberg table

| h | T (order h²) | Order h⁴ | Order h⁶ | Order h⁸ |
|---|---|---|---|---|
| 2 | 2.6597 |  |  |  |
| 1 | 3.1097 | 3.2597 |  |  |
| 0.5 | 3.2491 | 3.2955 | 3.2979 |  |
| 0.25 | 3.2804 | 3.2909 | 3.2906 | 3.2905 |

Sample extrapolation: R1 = (4 × 3.1097 − 2.6597)/3 = 3.2597; R2 = (16 × 3.2955 − 3.2597)/15 = 3.2979; R3 = (64 × 3.2906 − 3.2979)/63 = 3.2905.

The last two extrapolated values 3.2906 and 3.2905 agree to two decimal places (3.29), so the process stops.

**Answer: ∫₀² (eˣ + sin x)/(1 + x²) dx ≈ 3.29**

## Check yourself

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

1. What does Romberg's method do?
   A. Use Simpson only
   B. Fit a spline
   C. Differentiate
   D. Richardson extrapolation of trapezoidal estimates
   Answer: D. It improves T(h) estimates by extrapolation. (Key point: Idea)

2. Which is the first extrapolation formula?
   A. (4T(h/2) - T(h))/3
   B. T(h)/4
   C. (2T(h/2) - T(h))
   D. (T(h) + T(h/2))/2
   Answer: A. R = (4·finer - coarser)/3, e.g. 3.2597. (Key point: Formula)

3. Why stop at 3.29?
   A. h = 0
   B. Error is zero
   C. Last two extrapolated values agree to two decimals
   D. Table complete
   Answer: C. 3.2906 and 3.2905 agree to 3.29. (Key point: Stopping)

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