Antiderivatif

Antiderivatif, also known as an antiderivative or primitive function, is a function F whose derivative equals a given function f: F'(x) = f(x) for all x in the domain. If F is an antiderivative of f, then every other antiderivative of f has the form F(x) + C, where C is a constant. The indefinite integral, written as ∫ f(x) dx, denotes the process of finding such a function up to an additive constant.

Existence and interpretation: If f is continuous on an interval I, then f has at least one antiderivative on I, as guaranteed by the fundamental theorem of calculus. Antiderivatives provide a way to reverse differentiation. They are closely related to definite integrals, since ∫_a^b f(x) dx = F(b) − F(a) whenever F' = f.

Examples and common rules: The antiderivative of 2x is x^2 + C, of sin x is −cos x + C, and of e^x is e^x + C. More generally, for n ≠ −1, the antiderivative of x^n is x^{n+1}/(n+1) + C, and the antiderivative of 1/x is ln|x| + C. Antiderivatives obey linearity: ∫ (a f(x) + b g(x)) dx = a ∫ f(x) dx + b ∫ g(x) dx + C.

Limitations and extensions: Not all functions have antiderivatives that can be expressed with elementary functions. Some require special functions (for example, ∫ e^{−x^2} dx leads to the error function) or numerical methods for approximation.

Applications: Antiderivatives are used to compute areas, accumulated quantities, and to relate rates of change to total changes in physics, economics, biology, and engineering. They provide the reverse operation to differentiation and form a fundamental part of integral calculus.