Integral Calculator

How to Use the Integral Calculator

Enter your function

Type f(x) using ^ for powers and names like sin, cos, exp and 1/x — for example 3*x^2 + 2*x. You can write 2x for 2 times x.

Read the antiderivative

The calculator returns the indefinite integral — the antiderivative — with the constant of integration + C, the family of functions whose derivative is your input.

Add limits for a definite integral

Fill in "From" and "To" to evaluate the definite integral between those bounds — the signed area under the curve. You can use pi, e, or simple values like pi/2.

Exact or numerical

When an antiderivative exists the definite result is exact, by the Fundamental Theorem of Calculus. When it doesn't, the calculator still gives a precise numerical answer using Simpson's rule.

Integration, the Other Half of Calculus

Antiderivatives and Area

Integration is differentiation run in reverse, and it answers two questions that turn out to be the same question. The first is "what function has this as its derivative?" — its answer is the antiderivative, or indefinite integral. The second is "what is the area under this curve between two points?" — its answer is the definite integral. The astonishing discovery at the heart of calculus, the Fundamental Theorem, is that these are intimately linked: to find the area under a curve from a to b, you find an antiderivative and subtract its value at a from its value at b. That single insight, that area can be computed by anti-differentiation, is what makes calculus so powerful, and it is exactly what this calculator does. Type a function and it returns the antiderivative with the constant of integration; add bounds and it returns the area, computed exactly through that theorem whenever it can.

Like the companion derivative tool, this calculator does its work entirely in your browser with a small symbolic engine — it parses your function into a tree, applies the rules of integration (the reverse power rule, linearity, and the known antiderivatives of sine, cosine, the exponential and one-over-x), and simplifies the result. But integration has a famous difficulty that differentiation does not: not every function has an antiderivative that can be written down with elementary functions. The bell curve, e to the minus x squared, is the classic example — its antiderivative simply has no formula in terms of ordinary functions. When the calculator meets such a function it does not give up. For a definite integral it switches to a numerical method, Simpson's rule, which approximates the area by fitting parabolas to the curve and summing them, giving an answer accurate to many decimal places. So you always get a result: an exact formula when one exists, and a precise number when it does not.

"The Fundamental Theorem of Calculus says area and slope are inverse operations — to measure the area under a curve, undo the act of finding its slope."

Where Integrals Earn Their Keep

The definite integral — the area under a curve — is one of the most widely used ideas in quantitative work. Integrate velocity over time and you get distance travelled; integrate a rate of flow and you get the total volume; integrate a probability density and you get a probability. In engineering, integrals give the centre of mass of a shape, the work done by a varying force, and the heat accumulated over a process. In economics they turn a marginal cost curve back into total cost, and in statistics the area under the normal curve gives the chance of an outcome — the very integral with no elementary formula, which is why it is always computed numerically, exactly as this tool does. The constant of integration in the indefinite case is not a technicality either: it represents the fact that infinitely many functions share the same derivative, differing only by a starting value, and pinning it down is how initial conditions enter every differential-equation model of the physical world. Whether you are checking calculus homework, computing an area, or exploring a function that has no neat antiderivative, this calculator gives you the answer instantly and privately.

10 Facts About Integration

01

An integral measures the area under a curve — the accumulation of a quantity.

02

Integration is the reverse of differentiation — finding the antiderivative.

03

The Fundamental Theorem links area to antiderivatives: ∫ₐᵇ f = F(b) − F(a).

04

The + C reflects that many functions share one derivative.

05

Some functions, like e^(−x²), have no elementary antiderivative.

06

Simpson's rule approximates area by fitting parabolas to the curve.

07

Integrating velocity over time gives distance travelled.

08

The ∫ symbol is a stretched S, for "summa" — a sum of infinitely thin slices.

09

The area under the normal curve is a definite integral computed numerically.

10

Everything here runs in your browser — exact or numerical, nothing uploaded.

Frequently Asked Questions

An indefinite integral is the antiderivative — a function (plus a constant C) whose derivative is your input. A definite integral has bounds and gives a single number: the signed area under the curve between those two points. Leave the bounds blank for the antiderivative, or fill them in for the area.
Because differentiating a constant gives zero, infinitely many functions share the same derivative — they differ only by a constant. The + C stands for that whole family. When you supply bounds for a definite integral, the constant cancels out and the answer is a single number.
Some functions, like e^(−x²), genuinely have no antiderivative expressible with elementary functions. The calculator will tell you so — but if you provide bounds, it still evaluates the definite integral numerically with Simpson's rule, giving a precise decimal answer.
Type them into the "From" and "To" boxes. You can use plain numbers, the constants pi and e, or simple expressions such as pi/2. Both boxes must be filled for the definite integral to be evaluated.
When an antiderivative exists, yes — the calculator uses the Fundamental Theorem of Calculus to get the exact value, and labels it so. When no antiderivative formula exists, it labels the answer as numerical (Simpson's rule), which is accurate to many decimal places but is an approximation.
Polynomials and powers (the reverse power rule), sums and differences, constant multiples, and the standard antiderivatives of sine, cosine, the exponential, and one-over-x. More elaborate forms needing substitution or integration by parts may return "no elementary antiderivative", in which case a definite integral can still be evaluated numerically.
A definite integral measures signed area: parts of the curve below the horizontal axis count as negative. So a negative result means the curve spent more "area" below the axis than above it over that interval. It is entirely normal and correct.
No. Both the symbolic integration and the numerical Simpson's-rule evaluation happen entirely in your browser with a small built-in engine. Nothing is uploaded, and the tool works offline once the page has loaded.
Yes — change the "Variable" box to any letter, such as t. Every other letter is then treated as a constant, exactly as in single-variable integration.
Completely free, with no account or sign-up and no usage limit. It runs entirely in your browser and collects no data. Integrate as many functions as you like.

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