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定積分計算機

The definite integral of any function f(x) between two limits, by adaptive Gauss–Kronrod quadrature to about 20 significant digits, with the area shaded.

更新日 検証済みの例：7

Use x as the variable, e.g. x^2 * sin(x), 1/x, sqrt(x). Trig functions use radians here.
Numbers or expressions like pi/2
試す
積分
積分: 0.746824132812
小数点以下の最大桁数：12；最も近い値へ、等距離ならゼロから遠い値へ
Estimated error
1.1 × 10⁻²¹
Function evaluations
45
Subintervals used
2

The signed area between f(x) and the x-axis from 0 to 1 is 0.7468241328; area below the axis counts as negative. It took 45 evaluations over 2 subintervals.

Area under f(x) from 0 to 1

00.5100.250.50.751xf(x)ab
Subintervals 行数：2
変換元変換先ContributionError estimate
00.50.46133.2 × 10⁻²²
0.510.28558 × 10⁻²²
計算方法 S

  1. Scale each piece to [−1, 1]

    ∫αβf(x) dx=β−α2∫−11f ⁣(α+β2+β−α2 t)dt\int_{\alpha}^{\beta} f(x)\,dx = \frac{\beta-\alpha}{2}\int_{-1}^{1} f\!\left(\tfrac{\alpha+\beta}{2} + \tfrac{\beta-\alpha}{2}\,t\right)dt

  2. Kronrod 15-point rule with its embedded Gauss 7-point rule

    K15=β−α2∑i=115wif(xi),G7=β−α2∑j=17vjf(x2j),ε=ρ min⁡ ⁣(1,(200 ∣K15−G7∣ρ)3/2)K_{15} = \frac{\beta-\alpha}{2}\sum_{i=1}^{15} w_i f(x_i),\qquad G_7 = \frac{\beta-\alpha}{2}\sum_{j=1}^{7} v_j f(x_{2j}),\qquad \varepsilon = \rho\,\min\!\left(1, \left(\frac{200\,|K_{15} - G_7|}{\rho}\right)^{3/2}\right)

    The 7 Gauss nodes are reused by the 15-point rule, so the error estimate costs no extra evaluations; ρ is ∫|f − mean| over the piece (QUADPACK's scaling). Nodes and weights are carried to 52 digits.

  3. Adaptive bisection

    ∑ε=1.13×10−21≤max⁡(10−40, 10−20 ∣I∣)\sum \varepsilon = 1.13 \times 10^{-21} \le \max(10^{-40},\ 10^{-20}\,|I|)

    Split the subinterval with the largest error estimate until the total met the tolerance: 2 subintervals, 45 evaluations of f.

  4. 結果

    ∫01f(x) dx≈0.7468241328124270253994674\int_{0}^{1} f(x)\,dx \approx 0.7468241328124270253994674

定積分計算機について

A definite integral ∫ₐᵇ f(x) dx is the signed area between f and the x-axis from a to b, with area below the axis counted as negative. The calculator uses the 15-point Gauss–Kronrod rule from QUADPACK (routine QK15), which samples f at 15 points on each piece and compares the result with the embedded 7-point Gauss rule to estimate its own error. The piece with the largest error is split in two until the total estimated error falls below 10⁻²⁰ of the answer.

It suits integrals with no convenient antiderivative, such as the bell curve e^(−x²), and checking answers found by hand. The default, ∫₀¹ e^(−x²) dx, equals (√π/2)·erf(1) ≈ 0.7468241328 and took 45 evaluations of f over 2 subintervals.

Trig functions use radians, and limits accept expressions such as pi/2. Integrable endpoint singularities, like √x or ln x at 0, are handled because Gauss–Kronrod nodes never touch the endpoints. Swapping the limits flips the sign.

計算例

∫₀¹ e^(−x²) dx

Function f(x)
exp(-x^2)
Lower limit a
0
Upper limit b
1
積分
0.746824132812

照合元：(√π/2)·erf(1): Python decimal, erf by its Maclaurin series at 70 digits (hp.py)

∫₀^π sin x dx

Function f(x)
sin(x)
Lower limit a
0
Upper limit b
pi
積分
2

照合元：−cos π + cos 0 = 2 (antiderivative)

∫₁² 1/x dx

Function f(x)
1/x
Lower limit a
1
Upper limit b
2
積分
0.69314718056

照合元：ln 2: Python Decimal(2).ln()

∫₀³ x⁵eˣ dx

Function f(x)
x^5 * exp(x)
Lower limit a
0
Upper limit b
3
積分
1,686.671880008638

照合元：Antiderivative eˣ(x⁵ − 5x⁴ + 20x³ − 60x² + 120x − 120) evaluated in Python decimal

よくある質問

What does a definite integral represent?

The signed area between the curve and the x-axis over an interval: area above the axis counts as positive and area below as negative. ∫₀^π sin x dx = 2 because one arch of the sine curve encloses an area of 2, while ∫₀^2π sin x dx = 0 because the second arch lies below the axis and cancels the first. It is also a total change: integrating speed over time gives distance traveled.

How do you evaluate a definite integral by hand?

Find an antiderivative F with F′ = f, then subtract: ∫ₐᵇ f(x) dx = F(b) − F(a), which is the fundamental theorem of calculus. For ∫₁² 1/x dx, F(x) = ln x, so the answer is ln 2 − ln 1 ≈ 0.693147. When no elementary antiderivative exists, as for e^(−x²), a numerical rule like this one is the practical route.

What is Gauss–Kronrod quadrature?

A rule that estimates an integral from a weighted sum of f at carefully placed points. The 7-point Gauss rule integrates every polynomial up to degree 13 exactly; Kronrod's extension adds 8 more points and reuses the first 7, so comparing the two estimates gives an error bound without extra evaluations. QUADPACK's QK15 routine, used here, pairs it with repeated splitting of the worst subinterval.

What is the integral of e^(−x²)?

It has no antiderivative among elementary functions, so it is written with the error function: ∫₀ˣ e^(−t²) dt = (√π/2) erf(x). From 0 to 1 the value is about 0.7468241328, and over the whole real line it is √π ≈ 1.7724539. This Gaussian integral is what makes the normal distribution's total probability equal 1.

Can a definite integral be negative?

Yes, when more of the area lies below the x-axis than above it, or when the limits run backward. ∫₁² (−x) dx = −1.5, and the integral of x² from 1 down to 0 is −1/3, the negative of ∫₀¹ x² dx = 1/3. For the total area regardless of sign, integrate the absolute value instead: ∫₀^2π |sin x| dx = 4, entered as abs(sin(x)).

「定積分計算機」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：7。 例えば、「∫₀¹ e^(−x²) dx」は(√π/2)·erf(1): Python decimal, erf by its Maclaurin series at 70 digits (hp.py)と照合しています。

この計算方法の出典は何ですか？

Piessens, de Doncker-Kapenga, Überhuber, Kahaner — QUADPACK (1983), routine QK15; Wolfram MathWorld — Gauss-Kronrod Quadrature; Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997).

この計算機について

∫abf(x) dx≈b−a2∑i=115wi f(ti)ti=a+b2+b−a2 xi\begin{aligned} \int_a^b f(x)\,dx &\approx \frac{b-a}{2}\sum_{i=1}^{15} w_i\, f(t_i) \\[4pt] t_i &= \frac{a+b}{2} + \frac{b-a}{2}\,x_i \end{aligned}

出典

  1. Piessens, de Doncker-Kapenga, Überhuber, Kahaner — QUADPACK (1983), routine QK15
  2. Wolfram MathWorld — Gauss-Kronrod Quadrature
  3. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997)

出典と照合済み

この計算機には、独立した出典の解答を使った計算例が 7 件あります。テストに組み込まれており、ここでも実行できます。

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