What functions can I enter?

You can use any JavaScript Math expression with x as the variable: polynomials (x^3 + 2*x), trig (sin(x), cos(x), tan(x)), exponentials (e^x written as Math.exp(x) or exp(x)), logarithms (log(x) or ln(x)), and more. Powers use ^ or ** notation.

How accurate is the result?

Composite Simpson's rule with 1000 sub-intervals achieves roughly 8–10 decimal digits of accuracy for smooth functions. Error scales as O(h^4) where h = (b−a)/n, so increasing n dramatically improves precision.

What if the function has a discontinuity inside [a, b]?

Singularities or discontinuities inside the interval will degrade accuracy or produce NaN/Infinity. Split the integral at the discontinuity and sum the parts, or use a different bound.

Why does my result show NaN?

NaN usually means the function evaluates to an undefined value (e.g. log of a negative number, division by zero). Check the function syntax and ensure the domain is valid over the entire interval.

What is Simpson's rule?

Simpson's 1/3 rule approximates the integral by fitting parabolic arcs through consecutive triplets of points. The composite version divides [a, b] into n (even) sub-intervals and sums the weighted function values, giving error proportional to (b−a)^5 / (180 n^4).

Definite Integral Calculator

Name: Definite Integral Calculator
Availability: InStock
Author: Nham Vu

Enter a function and interval to compute its definite integral numerically using Simpson's rule.

Enter Function & Bounds

f(x) — function to integrate

Use x as the variable. Operators: + - * / ^ **. Functions: sin cos tan exp log sqrt abs.

Lower bound (a)

Upper bound (b)

Sub-intervals (n) — must be even

Result

Enter a function and click Calculate.

Common Examples

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Summary

Enter a function and interval to compute its definite integral numerically using Simpson's rule.

How it works

Enter the function f(x) using standard math notation (e.g. x^2, sin(x), e^x).
Set the lower bound (a) and upper bound (b) of the integration interval.
Choose the number of sub-intervals (higher = more accurate; default 1000).
Click "Calculate" to apply composite Simpson's 1/3 rule.
The numeric result is displayed with up to 10 significant digits.
Use the examples panel to load common integrals instantly.

Use cases

Check hand-computed definite integrals for calculus homework.
Estimate the area under a curve when an antiderivative is hard to find.
Verify results from symbolic CAS tools.
Compute integrals of physics and engineering functions (e.g. Gaussian, sinc).
Explore how integral value changes as bounds are adjusted.
Teach or demonstrate numerical integration methods in the classroom.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu