Integraale

Integraale, in the sense of the Romanian mathematical terminology, are central objects in calculus that describe accumulation and aggregation of quantities. They can be categorized as indefinite (or antiderivatives) and definite integrals, reflecting different ways of summing infinitesimal contributions over an interval or across a domain.

An indefinite integral, denoted by ∫ f(x) dx, represents the family of all antiderivatives of a function f, i.e., functions F whose derivative is f (F' = f). A definite integral, written as ∫_a^b f(x) dx, computes the net accumulation of f over the interval [a, b], and geometrically corresponds to the area between the graph of f and the x-axis in appropriate cases. The fundamental theorem of calculus links these two notions: if f is continuous on [a, b], then the function F(x) = ∫_a^x f(t) dt is an antiderivative of f, and ∫_a^b f(x) dx = F(b) − F(a).

Key properties include linearity, which allows the integration of sums and scaled functions, and common techniques such as substitution (u-substitution) and integration by parts. The study of integrals also distinguishes between elementary antiderivatives and cases where an antiderivative exists only in non-elementary form. Improper integrals extend the concept to unbounded intervals or integrands with infinite discontinuities.

Applications are widespread, including computation of areas, physical work, probability distributions, and accumulated quantities in economics and biology. In analysis, integrals have led to broader theories such as the Lebesgue integral, which generalizes the notion of area and accumulation beyond the Riemann framework. The development of integrals dates to the 17th century with contributions by Newton and Leibniz, and was further refined by Riemann and later by Lebesgue.