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Antiderivative ○꠹｜Definition｜1st｜20260604132833-00-⌔
Antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral1 of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as *F’ * = f.23 The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration).4 The discrete equivalent of the notion of antiderivative is antidifference.
Printed 2026-06-28.
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Antiderivatives are also called general integrals, and sometimes integrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to definite integrals. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it refers to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. This article adopts the latter approach. In English A-Level Mathematics textbooks one can find the term complete primitive - L. Bostock and S. Chandler (1978) Pure Mathematics 1; The solution of a differential equation including the arbitrary constant is called the general solution (or sometimes the complete primitive). ↩
Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. ↩
Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2. ↩
“4.9: Antiderivatives”. Mathematics LibreTexts. 2017-04-27. Retrieved 2020-08-18. ↩
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