antiderivata

An antiderivata, or antiderivative, of a function f is another function F defined on an interval such that F'(x) = f(x) for all x in that interval. It is the inverse operation of differentiation: finding a function whose derivative is f. The antiderivative is not unique: any F(x) + C, where C is a constant, is also an antiderivative. The process is represented by the indefinite integral ∫ f(x) dx, which denotes the family of all antiderivatives of f.

The Fundamental Theorem of Calculus links antiderivatives with definite integrals: if f is continuous on [a,b], then the function F(x) = ∫_a^x f(t) dt is an antiderivative of f on [a,b], and ∫_a^b f(x) dx = F(b) − F(a).

Examples: The antiderivative of x^n with n ≠ −1 is x^{n+1}/(n+1) + C. The antiderivative of e^x is e^x + C. The antiderivative of 1/x is ln|x| + C on any interval not containing 0.

Existence and variety: Every continuous function on an interval has an antiderivative, but not every antiderivative can be expressed with elementary functions; some require special functions or numerical methods.

Applications: Antiderivatives are used to compute areas under curves, accumulate quantities over an interval, and solve certain differential equations.