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Integral Calculator

Evaluate a definite integral numerically with Simpson’s rule, and see the antiderivative where a closed form exists.

Written by Jude Ramsey · Last updated August 2026 · How these calculators are checked

How integral is worked out

The definite integral is computed by Simpson’s rule, which fits parabolas through successive triples of points and is accurate to several decimal places for smooth functions. This works even when no closed-form antiderivative exists.

Where the integrand fits a recognisable pattern β€” polynomials, sine, cosine, exponentials, 1/x and square roots β€” the calculator also reports the symbolic antiderivative and evaluates it at both limits so you can compare methods.

A worked example

As an example, with a function f of x^2 + 3x, numeric method of Simpson’s rule β€” 2,000 intervals, the calculator returns a definite integral of 45.333333. Underneath, Antiderivative F comes out at x^3/3+3Β·x^2/2 + C and F at 45.333333. Change any field and every figure updates as you type.

How to use it

  1. Enter the integrand as a function of x.
  2. Set the lower and upper limits.
  3. Choose how many intervals Simpson’s rule should use.

How the number is calculated

Simpson’s rule: ∫ β‰ˆ (h/3)[fβ‚€ + 4f₁ + 2fβ‚‚ + … + fβ‚™] fundamental theorem: βˆ«β‚α΅‡ f = F(b) βˆ’ F(a)

Frequently asked questions

Why is the numeric method the main result?
Because it always produces an answer. Symbolic antidifferentiation fails for many perfectly ordinary functions, such as e^(βˆ’xΒ²).
How accurate is Simpson’s rule?
Very, for smooth functions β€” typically ten or more correct digits with a few thousand intervals. Accuracy suffers near singularities.
What does the “+ C” mean?
An indefinite integral is only determined up to an additive constant, since the derivative of any constant is zero.

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