Numerical Integration Calculator
Simpson’s rule fits parabolas through the ordinates: ∫₀¹ 1/(1 + x²) dx with 6 intervals gives 0.785397945234, against π/4 = 0.785398.
Use x, ^ for powers, and sin, cos, tan, exp, ln, log, sqrt, abs, pi, e. 2x and xsin(x) are understood.
Simpson’s 1/3 needs an even n; 3/8 needs a multiple of 3.
0.785397945234011
Full answer, then cut to fewer decimals (not rounded):
- 0.785397945234011
- 0.78539794523401
- 0.7853979452340
- 0.785397945234
- 0.78539794523
- 0.7853979452
- 0.785397945
- 0.78539794
- 0.7853979
- 0.785397
- 0.78539
- 0.7853
- 0.785
- 0.78
- 0.7
- Trapezoidal rule
- 0.7842407666
- Simpson’s 1/3 rule
- 0.7853979452
- Simpson’s 3/8 rule
- 0.7853958624
- Reference value (adaptive)
- 0.7853981634
- Step h = (b − a)/n
- 0.1666666667
| i | xᵢ | f(xᵢ) |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 0.16666667 | 0.972972973 |
| 2 | 0.33333333 | 0.9 |
| 3 | 0.5 | 0.8 |
| 4 | 0.66666667 | 0.6923076923 |
| 5 | 0.83333333 | 0.5901639344 |
| 6 | 1 | 0.5 |
Step-by-step solution
- h = (1 − 0) ÷ 6 = 0.1666666667
- Trapezoidal: h/2 [y₀ + 2(y₁ + … + yₙ₋₁) + yₙ] = 0.7842407666
- Simpson 1/3: h/3 [y₀ + 4(odd) + 2(even) + yₙ] = 0.7853979452
Numerical Integration Formula
Simpson: h/3 [y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]
trapezoidal: h/2 [y₀ + 2(y₁ + … + yₙ₋₁) + yₙ]
Worked Example
∫₀¹ 1/(1 + x²) dx with n = 6:
- h = (1 − 0) ÷ 6 = 0.1666666667
- Trapezoidal: h/2 [y₀ + 2(y₁ + … + yₙ₋₁) + yₙ] = 0.7842407666
- Simpson 1/3: h/3 [y₀ + 4(odd) + 2(even) + yₙ] = 0.7853979452
Simpson’s 1/3 rule: 0.7853979452 · Simpson’s 3/8 rule: 0.7853958624 · Reference value (adaptive): 0.7853981634 · Step h = (b − a)/n: 0.1666666667
| i | xᵢ | f(xᵢ) |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 0.16666667 | 0.972972973 |
| 2 | 0.33333333 | 0.9 |
| 3 | 0.5 | 0.8 |
| 4 | 0.66666667 | 0.6923076923 |
| 5 | 0.83333333 | 0.5901639344 |
| 6 | 1 | 0.5 |
GATE tip
Check n before choosing a rule: Simpson 1/3 needs an even n and 3/8 a multiple of 3. With n = 6 both work and give a quick cross-check.
More Numerical Methods Calculators
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Numerical Differentiation Calculator
f′(x) ≈ [f(x + h) − f(x − h)] ÷ 2h
forward and backward differences have O(h) error; central O(h²)
Example: f(x) = sin x at x = 1 with h = 0.1: Central difference f′(x) ≈ 0.53940225217.
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Newton-Raphson Method Calculator
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Example: f(x) = x³ − 2x − 5 starting at x₀ = 2: Root x ≈ 2.09455148154.
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Bisection Method Calculator
c = (a + b)/2; regula falsi c = (a·f(b) − b·f(a)) ÷ (f(b) − f(a))
keep the half where f changes sign
Example: f(x) = x³ − x − 2 on [1, 2]: Root x ≈ 1.5213804245.
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Secant Method Calculator
xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁) ÷ (f(xₙ) − f(xₙ₋₁))
order of convergence ≈ 1.618
Example: f(x) = x² − 612 starting from 10 and 30: Root x ≈ 24.7386337537.
Frequently asked questions
What is the error of the trapezoidal and Simpson rules?
Trapezoidal error ∝ h² and is exact for straight lines; Simpson’s 1/3 error ∝ h⁴ and is exact for cubics.
What if I only have a table of values?
Use the Simpson’s rule calculator for tabulated ordinates; this page samples f(x) for you.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Numerical Integration Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/numerical-integration-calculator/
IEEE
Scientific Converters, "Numerical Integration Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/numerical-integration-calculator/