antiderivée

In calculus, an antiderivative of a function f is another function F such that F'(x) = f(x) for all x in an interval where f is defined. If such F exists, f is said to have a primitive on that interval.

The process of finding an antiderivative is called integration in the indefinite sense. The indefinite integral is written ∫ f(x) dx, and it denotes the family of all antiderivatives of f. Any two antiderivatives differ by a constant, so ∫ f(x) dx = F(x) + C.

Fundamental theorem of calculus: If f is continuous on [a,b], the function F(x) = ∫_a^x f(t) dt is an antiderivative of f, and ∫_a^b f(x) dx = F(b) − F(a). This theorem links differentiation and integration, showing that differentiation and integration are inverse processes.

Examples: The antiderivative of 2x is x^2 + C, since (x^2)' = 2x. The constant function 0 has antiderivatives equal to any constant. More generally, polynomials have polynomial antiderivatives, obtained by increasing the exponent and dividing by the new exponent.

Domain and constants: An antiderivative is defined on an interval where f is defined; different intervals can yield different antiderivatives. The constant of integration C reflects the family of primitive functions.

Existence and limitations: Every continuous function on an interval has an antiderivative on that interval. Some functions do not have antiderivatives that can be expressed with elementary functions, requiring numerical methods or special functions.