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정적분 계산기

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.

업데이트 검증한 예제: 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
시도하기
적분
적분: 0.746824132812
최대 소수 자릿수: 12; 가장 가까운 값, 중간값은 0에서 먼 쪽으로
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 행 수: 2
변환 전변환 후ContributionError estimate
00.50.46133.2 × 10⁻²²
0.510.28558 × 10⁻²²
계산 방법 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. 결과

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

정적분 계산기 소개

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.

계산 예제

∫₀¹ e^(−x²) dx

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

검증 출처: (√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
적분
2

검증 출처: −cos π + cos 0 = 2 (antiderivative)

∫₁² 1/x dx

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

검증 출처: ln 2: Python Decimal(2).ln()

∫₀³ x⁵eˣ dx

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

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

자주 묻는 질문

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)).

“정적분 계산기”의 정확도는 어느 정도인가요?

정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 7. 예를 들어 “∫₀¹ e^(−x²) dx”은 (√π/2)·erf(1): Python decimal, erf by its Maclaurin series at 70 digits (hp.py)와 대조해 확인합니다.

이 계산 방법의 출처는 무엇인가요?

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).

이 계산기 소개

∫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}

출처

  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)

출처와 대조하여 검증

이 계산기에는 독립적인 출처에서 답을 얻은 계산 예제가 7개 있습니다. 테스트 모음에서 실행되며 여기에서도 실행할 수 있습니다.

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