What Is an Integral Calculator?
An integral calculator computes the integral of a function — either an indefinite integral (the antiderivative) or a definite integral (a numerical value over an interval). Integration is one of the two pillars of calculus, used to find areas under curves, accumulated quantities, volumes, work, and probabilities. Enter a function and the calculator returns its integral, and for definite integrals it evaluates the result between your chosen limits.
How to Use the Integral Calculator
- Enter a function — for example x^2, sin(x), or e^x.
- Choose integral type — indefinite (antiderivative) or definite (with limits).
- For definite integrals, enter the lower and upper bounds.
- Calculate — see the integral and, where available, the steps.
Indefinite vs Definite Integrals
An indefinite integral finds a function whose derivative is the integrand, written with a constant of integration: ∫ f(x) dx = F(x) + C. A definite integral evaluates the antiderivative between two limits to produce a single number, representing the signed area under the curve: ∫ₐᵇ f(x) dx = F(b) − F(a).
Common Integration Rules
| Function | Integral |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ ÷ (n + 1) + C |
| 1 ÷ x | ln|x| + C |
| eˣ | eˣ + C |
| sin(x) | −cos(x) + C |
| cos(x) | sin(x) + C |
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration: 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). This is why finding an antiderivative lets you compute exact areas and accumulated change.
Where Integration Is Used
- Geometry: area under curves and between curves, volumes of revolution.
- Physics: distance from velocity, work from force, center of mass.
- Probability: areas under probability density functions.
- Engineering and economics: accumulated quantities and consumer surplus.
Frequently Asked Questions
How do you solve an integral?
Find a function whose derivative equals the integrand, applying rules like the power rule and substitution. The calculator does this and, for definite integrals, evaluates between the limits.
What is the difference between a definite and indefinite integral?
An indefinite integral gives a general antiderivative with a constant C, while a definite integral evaluates that antiderivative between two limits to produce a specific number.
What is an antiderivative?
An antiderivative of a function f is a function F whose derivative is f. Integration is the process of finding antiderivatives.
Why is there a +C in indefinite integrals?
Because the derivative of a constant is zero, infinitely many functions share the same derivative. The constant of integration C accounts for all of them.
Is this integral calculator free?
Yes — it is completely free, requires no signup, and handles both definite and indefinite integrals.