Integrale

An integral, or integrale in some languages, is a central concept in calculus that formalizes accumulation. It can be viewed as the limit of sums of infinitesimal quantities, and as the measure of the area, length, or other quantities accumulated along a domain. There are two main kinds: the indefinite integral, which represents a family of antiderivatives, and the definite integral, which evaluates to a single number over an interval.

Notation typically uses the symbol ∫. The indefinite integral is written ∫ f(x) dx and represents all functions F whose derivative is f, i.e., F' = f. The definite integral ∫_a^b f(x) dx computes the net accumulation of f from a to b; its value depends on the behavior of f on the interval and is independent of the particular variable of integration.

One of the cornerstones is the Fundamental Theorem of Calculus, which links differentiation and integration. If f is continuous on [a,b] and F is defined by F(x) = ∫_a^x f(t) dt, then F is differentiable on (a,b) and F' = f. The theorem also implies that definite integrals can be evaluated using antiderivatives: ∫_a^b f(x) dx = F(b) - F(a).

Many integrals can be computed analytically using techniques such as substitution, integration by parts, partial fractions, and trigonometric or special substitutions. Others require numerical methods, including the trapezoidal rule, Simpson's rule, or more sophisticated algorithms, especially for improper or non-elementary integrals.

Beyond the Riemann integral, broader notions include the Lebesgue integral, suitable for more general functions; and improper integrals, defined as limits of integrals on unbounded intervals or near singularities. Integrals have wide applications in physics, geometry, statistics, and economics.