Integral Calculator

A definite integral computes the exact signed area between a function f(x) and the x-axis over the interval [a, b]. When an exact antiderivative is known, the Fundamental Theorem of Calculus gives the exact answer. When it isn’t, numerical approximation methods are used.

Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a) where F’(x) = f(x)

Common antiderivative rules:

• ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (power rule, n ≠ −1)
• ∫ eˣ dx = eˣ + C
• ∫ sin(x) dx = −cos(x) + C
• ∫ cos(x) dx = sin(x) + C
• ∫ 1/x dx = ln|x| + C

Numerical approximation — Simpson’s Rule: ∫ₐᵇ f(x) dx ≈ (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 4f(xₙ₋₁) + f(xₙ)] where n must be even, and h = (b − a) / n.

What each variable means:

• a, b: lower and upper bounds of integration
• f(x): the integrand (the function being integrated)
• F(x): the antiderivative of f(x)
• h: interval width = (b − a) / n
• n: number of subintervals (Simpson’s requires even n; more intervals = more accuracy)

Worked example — Exact: ∫₁⁴ (3x² − 2x + 1) dx Antiderivative F(x) = x³ − x² + x F(4) = 64 − 16 + 4 = 52 F(1) = 1 − 1 + 1 = 1 Result = 52 − 1 = 51

Worked example — Simpson’s Rule: ∫₀² eˣ dx using n = 4: h = 0.5; x values: 0, 0.5, 1.0, 1.5, 2.0 f values: 1.000, 1.649, 2.718, 4.482, 7.389 Sum = 1.000 + 4(1.649) + 2(2.718) + 4(4.482) + 7.389 = 1 + 6.596 + 5.436 + 17.928 + 7.389 = 38.349 Integral ≈ (0.5/3) × 38.349 = 6.3915 Exact: e² − e⁰ = 7.389 − 1 = 6.389 — Simpson’s is accurate to within 0.003!