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Romberg: (e^x + sin x)/(1+x²)

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.

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

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)

xf(x)
01.0000
0.251.4413
0.51.7025
0.751.7911
11.7799
1.251.7324
1.51.6859
1.751.6587
21.6597

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

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

hT (order h²)Order h⁴Order h⁶Order h⁸
22.6597
13.10973.2597
0.53.24913.29553.2979
0.253.28043.29093.29063.2905

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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