lintégration

Lintégration, here used to refer to the mathematical operation known in English as integration, is the process of determining an integral. It expresses accumulation, such as area under a curve, total quantity over an interval, or the antiderivative of a function. There are two primary senses: indefinite integrals, which represent families of antiderivatives F with F' = f, written ∫ f(x) dx = F(x) + C; and definite integrals, written ∫_a^b f(x) dx, which measure accumulation on the interval [a, b], often yielding a number such as an area or total mass.

The fundamental theorem of calculus links differentiation and integration: under suitable conditions, the integral of a function over [a, b] equals the net change of its antiderivative, F(b) − F(a). This provides a bridge between geometry and analysis.

Techniques for finding lintégration include substitutions, integration by parts, partial fraction decomposition, and special functions. When an antiderivative cannot be expressed in closed form, numerical methods such as the trapezoidal rule or Simpson’s rule estimate the value. Improper integrals extend the notion to unbounded intervals or unbounded integrands.

In broader mathematics, lintégration is formalized in different frameworks such as the Riemann integral and the Lebesgue integral, which generalize the notion of area and accumulate values in more complex settings. Applications span physics, engineering, probability, economics, and beyond, including work done by a force, probability distributions, and total resources over time.