Intégrales

Intégrales are a fundamental concept in calculus, representing the process of summation. They are broadly categorized into two main types: indefinite integrals and definite integrals. An indefinite integral, often denoted by ∫f(x) dx, represents the family of all antiderivatives of a function f(x). The antiderivative is a function whose derivative is f(x). The result of an indefinite integration is a function plus an arbitrary constant of integration, usually denoted by C. This constant arises because the derivative of a constant is zero.

Definite integrals, on the other hand, evaluate to a single numerical value. They are written as ∫[a,b] f(x) dx, where 'a' and 'b' are the lower and upper limits of integration, respectively. The definite integral represents the signed area between the curve of the function f(x) and the x-axis, over the interval [a,b]. The Fundamental Theorem of Calculus establishes a crucial link between differentiation and integration, stating that the definite integral of a function from a to b can be found by evaluating its antiderivative at b and subtracting its value at a. Intégrales have wide-ranging applications in physics, engineering, economics, and statistics, used for calculating areas, volumes, work, probability, and much more.