antiderivative

An antiderivative of a function f defined on an interval I is a function F such that F'(x) = f(x) for every x in I. In other words, F is a primitive of f. Antiderivatives are not unique: if F is an antiderivative of f, then so is F(x) + C for any constant C. The term antidifferentiation refers to the process of finding such a function, and the notation ∫ f(x) dx is used to denote the family of all antiderivatives of f, i.e., {F(x) + C}.

The Fundamental Theorem of Calculus links differentiation and integration. If f is continuous on an interval [a, b], the function F defined by F(x) = ∫_a^x f(t) dt is an antiderivative of f, and the definite integral equals F(b) − F(a). The theorem establishes that differentiation and integration are inverse processes in a precise sense and provides a practical method for evaluating certain integrals via antiderivatives.

Common antiderivatives include several standard rules: ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ -1; ∫ e^{ax} dx = (1/a) e^{ax} + C; ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; and ∫ (1/x) dx = ln|x| + C, for x ≠ 0. In practice, finding antiderivatives uses techniques such as substitution, integration by parts, and partial fractions.

Not all functions have antiderivatives that can be expressed in elementary functions. Some require special functions or numerical methods to evaluate. Nevertheless, every continuous function on an interval has at least one antiderivative, ensuring the existence of indefinite integrals in the broad sense.