Definite Integral Calculator

Evaluate definite and improper integrals: analytically when an antiderivative exists, otherwise by adaptive quadrature.

Your result will appear here

Fill in the fields and press Integrate, or pick one of the examples.

About the Definite Integral Calculator

Evaluates $\int_a^b f(x)\,dx$ . When a verified antiderivative exists, the value comes from the Fundamental Theorem of Calculus and is cross-checked numerically. Otherwise the value is computed by adaptive Gauss–Kronrod quadrature and labelled as a numerical approximation with an error estimate.

Limits may be expressions (pi/2) or infinite (inf, -inf). Improper integrals that do not converge are reported as such rather than given a meaningless number.

Limitations

“Exact” values are the antiderivative evaluated in floating point (12 significant digits); a closed form such as π/4 is not produced.
Integrands with non-integrable singularities inside the interval are rejected. Highly oscillatory integrands may fail to converge.

How to use it

Type the integrand.
Enter the lower and upper limits; use inf or -inf for infinity.
Press Integrate and check whether the result is marked Exact or Approximate.

Example

∫₀³ x^2 dx

→ 9

Formulas used

\int_a^b f(x)\,dx = F(b) - F(a),\quad F' = f

Fundamental Theorem of Calculus (f continuous on [a, b]).

Integral CalculatorIndefinite integrals (antiderivatives) found symbolically and verified by differentiation before they are shown.Numerical Integration CalculatorApproximate integrals with the trapezoidal, midpoint and Simpson’s rules and compare them with an adaptive reference value.Graphing CalculatorPlot and compare functions, implicit equations and inequalities with zoom, pan, coordinate tracking and PNG export.

About the Definite Integral Calculator

Limitations

How to use it

Formulas used

Related tools

About the Definite Integral Calculator

Limitations

How to use it

Formulas used