--- title: "Integral Calculator - Definite Integrals | CanDoYa" description: "Free integral calculator: enter a function and lower and upper limits to evaluate the definite integral instantly. No sign-up, no download - runs in your browser." url: https://candoya.com/tools/integral-calculator/ locale: en type: browser-tool --- # Free Integral Calculator Category: Calculators ## What is an integral calculator? An integral calculator evaluates the definite integral of a function over an interval - the signed area under its curve between a lower and an upper limit. Type a function of x, set the two limits, and this free tool returns the value of ∫ₐᵇ f(x) dx instantly, computed numerically in your browser with no sign-up. Everything runs locally: your function is parsed on the page, integrated with an adaptive numerical method, and nothing is uploaded or stored. Limits can be numbers or expressions like `pi`, `2*pi` or `1/2`, and the result updates as you type. This is a definite-integral calculator - it returns the numeric area for a chosen interval rather than a symbolic antiderivative. ## How to use it 1. **Enter the function** - Type f(x) using x as the variable - for example x^2 + sin(x). Use ^ for powers and functions like sin, cos, exp, ln and sqrt. 2. **Set the limits** - Enter the lower limit a and the upper limit b. Each can be a number or an expression such as pi, 2*pi or 1/2. 3. **Read the value** - The definite integral ∫ₐᵇ f(x) dx appears immediately. Use Copy result to grab it, or an example chip to try a common integral. ## Who it's for - **Calculus students** checking a definite integral from a homework problem or textbook without opening Wolfram Alpha or a graphing calculator. - **Engineers and physicists** evaluating the area under a curve - work, displacement, or a probability - straight from a formula. - **Data and finance analysts** integrating a modelled function over a range to get a total or an expected value. - **Teachers** demonstrating how the value of a definite integral changes as the limits move, with numbers a class can edit live. - **Anyone** who wants the numeric answer to ∫ₐᵇ f(x) dx fast, including improper cases like ∫₀¹ 1/√x dx that still converge. ## Expert note This tool computes definite integrals numerically with adaptive Simpson's rule, not symbolically - so it returns a highly accurate number, not a formula for the antiderivative. That matters in two ways. First, results carry a tiny rounding error (usually far below what you would round to anyway), and a function with a spike or singularity inside the interval is flagged as approximate rather than reported falsely as exact. Second, a genuinely divergent integral, such as ∫₋₁¹ 1/x dx, is reported as non-convergent instead of returning a misleading finite value - while an improper but convergent integral like ∫₀¹ ln(x) dx = -1 is still evaluated correctly. ## Frequently asked questions ### Is the integral calculator free? Yes - it is 100% free, with no sign-up and no limit on how many integrals you evaluate. The whole calculation runs in your browser and nothing you type is uploaded. ### Does it compute definite or indefinite integrals? It computes definite integrals: you give a lower and an upper limit and it returns the numeric value of the area under the curve between them. It does not return an indefinite integral (a symbolic antiderivative with + C). ### How does the calculator evaluate the integral? It parses your function into a fast evaluator and integrates it with adaptive Simpson's rule, a numerical method that subdivides the interval wherever the curve bends sharply until the result stops changing. This gives an answer accurate to about 8-10 significant figures for well-behaved functions. ### What functions and constants can I use? You can use +, -, *, / and ^, parentheses, the constants pi and e, and functions including sin, cos, tan, asin, acos, atan, exp, ln, log (base 10), sqrt, cbrt and abs. Implicit multiplication like 2x or 2sin(x) works too. ### Can the limits be pi or fractions? Yes. Each limit can be a number or an expression such as pi, 2*pi, pi/4 or 1/2. If the lower limit is larger than the upper limit, the sign of the result flips, exactly as the definition of the integral requires. ### Can it handle improper integrals? It handles many improper integrals that converge, such as ∫₀¹ 1/√x dx = 2, by carefully sampling near the singularity, and marks the result as approximate. If the integral diverges - for example ∫₋₁¹ 1/x dx - it tells you the integral does not converge instead of printing a wrong number. Infinite limits are not supported. ### How accurate is the result? For smooth functions the value is typically accurate to 8-10 significant figures, well beyond everyday needs. Near a singularity the tool falls back to a best-effort estimate and labels it approximate, so you always know when to treat the last digits with caution. ### Why does my result look slightly off from the exact answer? Because it is a numerical method, the answer can differ from an exact symbolic value in the final digit or two - for instance returning 0.6666666667 for 2/3. Round to the precision you need and the values agree. ## Related tools - [Correlation Coefficient Calculator](https://candoya.com/tools/correlation-coefficient/) - [Standard Deviation Calculator](https://candoya.com/tools/standard-deviation-calculator/) - [Percentage Calculator](https://candoya.com/tools/percentage-calculator/) --- *Markdown edition of , published for AI agents and other automated readers. Index of key pages: .*