Find an indefinite antiderivative or evaluate a definite integral between two bounds. Results include a simplified symbolic expression, an independent numeric check for definite integrals, and a graph of the function over the interval.
How to use this calculator
Enter the function and lower and upper bounds, then check that the function is defined throughout the interval. The result is signed area: portions below the axis subtract from portions above it. Numerical integration may be unreliable near discontinuities or singularities.
Examples
Integrate x²
The indefinite integral of x² is x³/3 + C. From 0 to 3, its definite integral is 9.
Integrate sin(x)
An antiderivative of sin(x) is −cos(x). From 0 to π, the definite integral is 2.
Formula
The Fundamental Theorem of Calculus connects antiderivatives and definite integrals:
if F′(x) = f(x), then ∫[a,b] f(x) dx = F(b) − F(a)
An indefinite result includes + C because all functions differing by a
constant have the same derivative.
Integral limits
An indefinite integral represents a family of antiderivatives and therefore includes an arbitrary constant. A definite integral is signed accumulation; it equals geometric area only after regions below the axis are handled appropriately. Improper integrals, singularities, oscillation, and branch choices may require convergence analysis beyond a displayed antiderivative.