Calculus 12 · Mathematics · Lesson 5.1
Antiderivatives
- Level
- Grade 12
- Reading time
- 13 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Reversing differentiation.
What is an antiderivative?
Short answer
An antiderivative of a function f is a function F whose derivative is f, that is F′(x) = f(x). Finding antiderivatives — also called indefinite integration — reverses differentiation. Because the derivative of a constant is zero, every function has infinitely many antiderivatives differing by a constant C, written ∫ f(x) dx = F(x) + C.
Concept preview
What you will be able to do
- 01Interpret an antiderivative as a function whose derivative is a given function.
- 02Determine antiderivatives of powers, sums and constant multiples using the reverse power rule.
- 03Apply the indefinite integral with an arbitrary constant of integration.
- 04Solve an initial-value problem to find one specific antiderivative from a condition.
- 05Diagnose a flawed antidifferentiation argument, such as omitting the constant of integration.
Every operation in mathematics invites its reverse, and differentiation is no exception. So far the derivative has answered the question, given a function, what is its rate of change; the reverse question asks, given a rate of change, what function has it. A function whose derivative is a given function is called an antiderivative of that function, and the process of finding one is antidifferentiation. This reversal is not a mere curiosity: it is the gateway to integration and to recovering a quantity from the rate at which it changes, such as finding position from velocity or total accumulated amount from a rate of accumulation. Learning to undo differentiation is therefore the first step into the second half of calculus, and it leans entirely on the differentiation rules you already know, run in reverse.
School Plan
Antiderivatives
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Related topics
Frequently asked questions
- Why do antiderivatives include "+ C"?
- Because the derivative of any constant is zero, so adding any constant to an antiderivative gives another valid one.
- How is an antiderivative related to a definite integral?
- The Fundamental Theorem of Calculus evaluates a definite integral using an antiderivative: ∫ₐᵇ f = F(b) − F(a).