∫Integral Calculator
Compute a definite integral with Simpson's rule, and see the symbolic antiderivative for polynomial and 1/x functions.
Calculations run entirely in your browser using math.js for parsing and evaluation, with Simpson's rule for the definite integral. No data is sent to a server.
Free Integral Calculator
Our free integral calculator computes the definite integral of a function over an interval [a, b] using Simpson's rule, a numerical method accurate to a very high number of decimal places for smooth functions. Enter any function — polynomial, trigonometric, exponential, or rational — along with the bounds, and get an instant result.
For polynomial functions and functions of the form 1/x, the calculator also derives the symbolic antiderivative using the power rule (∫xⁿ dx = xⁿ⁺¹/(n+1)) and the natural log rule (∫1/x dx = ln|x|), then evaluates it at both bounds using the Fundamental Theorem of Calculus to confirm the numeric result.
This makes the tool useful for checking calculus homework, verifying area-under-the-curve calculations, and understanding how a definite integral connects to its antiderivative. No sign-up, no downloads — everything runs instantly in your browser.
About the Integral Calculator
Features
Simpson's rule integration
Accurate numeric integration for any function math.js can parse.
Symbolic antiderivatives
Power rule and natural log rule antiderivatives for polynomial and 1/x functions.
Quick-start presets
Common example functions to load with one click.
Step-by-step working
See the antiderivative and the Fundamental Theorem of Calculus substitution when available.
Frequently Asked Questions
What is a definite integral?
A definite integral ∫ₐᵇ f(x)dx represents the signed area between a function's curve and the x-axis, between the bounds a and b.
What is Simpson's rule?
Simpson's rule is a numerical integration method that approximates a definite integral by fitting parabolas through groups of points, giving very accurate results even with a moderate number of subintervals.
When does the calculator show a symbolic antiderivative?
For polynomial functions (sums of terms like c·xⁿ) and functions of the form 1/x, the calculator derives the antiderivative using the power rule and the natural log rule. For other functions, only the numeric result is shown.
How accurate is the numeric result?
The calculator uses Simpson's rule with 2000 subintervals, which is accurate to many decimal places for smooth, continuous functions over the chosen interval.
What functions can I enter?
Any function math.js can parse, including polynomials, trigonometric functions, exponentials, logarithms, and rational functions like 1/x.
Can the bounds be in any order?
Yes. If a is greater than b, the calculator computes the integral over [b, a] and applies the sign flip, matching ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx.
What happens if the function is undefined in the interval?
If the function produces a non-finite value anywhere in the interval (such as 1/x at x=0), the calculator flags that the integral cannot be computed over that range.
What is the Fundamental Theorem of Calculus?
It states that if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). The calculator uses this to verify the numeric result when a symbolic antiderivative is available.
Can I check my homework with this calculator?
Yes, enter your function and bounds to instantly verify a definite integral, and compare against the symbolic antiderivative when available.
Is this calculator free to use?
Yes, the integral calculator is completely free with no sign-up, and all calculations run instantly in your browser.