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Definite Integral Calculator

Numerically integrate a function between two finite bounds. Check the approximate value and estimated numerical error.

Inputs
Enter values to calculate

Use * to multiply and a decimal point: 2*x, x^2, sin(x), cos(x), tan(x), exp(x), ln(x), sqrt(x), abs(x), pi, e. Angles are in radians.

Only functions continuous throughout the interval. Improper integrals, discontinuities and rapid oscillations are unsupported. No antiderivatives.

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Results

Enter a continuous function and integration bounds.

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How to calculate a definite integral

Enter a function of x and two finite bounds. Use explicit multiplication: 2*x, not 2x. Write powers as x^2; ln(x) and log(x) both mean the natural logarithm. Trigonometric functions use radians. Use a decimal point in the expression, such as 0.5*x.

Simpson integration approximates each section with a parabola and subdivides where more accuracy is needed. For one section, the rule is (b − a) × [f(a) + 4f((a + b)/2) + f(b)] / 6. This tool combines multiple sections and reports an estimate of numerical error.

Step-by-step example

For the initial values, f(x) = x² from 0 to 3:

  1. A known antiderivative is x³ / 3.
  2. Evaluate at the endpoints: 3³ / 3 − 0³ / 3.
  3. The answer is 9, a check on the tool's numerical result.

Other useful checks: sin(x) from 0 to pi is approximately 2; x from −1 to 1 gives 0 because signed areas cancel. Reversing the bounds of x² to 3 and 0 produces −9.

Method limitations

This tool does not perform symbolic integration. Do not use it for improper integrals such as 1/x from −1 to 1, discontinuous functions or rapid oscillations. A function may have a singularity between sampled points that the method misses. A small estimated error does not prove continuity or guarantee accuracy. Verify sensitive results with an independent analytical method.

Source

OpenStax: numerical integration describes Simpson's rule and conditions for error estimates. To find roots of second-degree polynomials, use the equation calculator.

Frequently asked questions

No. It approximates a definite integral with finite bounds using adaptive Simpson integration. It does not return an antiderivative or a constant C.

No. The integral is signed area: regions below the x-axis subtract. Reversing the bounds reverses the sign.