AP Calculus AB Flashcards: Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
In the expression ∫ f(x) dx = F(x) + C, what is the relationship between F(x) and f(x)?
F(x) is an antiderivative of f(x), which means that the derivative of F(x) is f(x), or F'(x) = f(x).
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In the expression ∫ f(x) dx = F(x) + C, what is the relationship between F(x) and f(x)?
F(x) is an antiderivative of f(x), which means that the derivative of F(x) is f(x), or F'(x) = f(x).
What is the general form of an indefinite integral for a function f(x)?
The indefinite integral is expressed as ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is any constant.
If you know that the derivative of sin(x) is cos(x), what is the indefinite integral of cos(x)?
Using the knowledge of derivatives, the indefinite integral is ∫ cos(x) dx = sin(x) + C.
Define an antiderivative of a function f.
An antiderivative of a function f is a function F whose derivative is f; that is, F'(x) = f(x).
What does the notation ∫ f(x) dx represent?
This notation represents the indefinite integral of the function f with respect to x, which is the family of all its antiderivatives.
How is the process of finding a derivative used to determine an antiderivative?
To find an antiderivative of a function, you must determine a new function which, when differentiated, results in the original function.
Is it possible to find a closed-form antiderivative for every function?
No, many functions do not have closed-form antiderivatives, meaning their antiderivative cannot be expressed in terms of elementary functions.
What provides the foundation for finding antiderivatives?
Differentiation rules provide the foundation for finding antiderivatives, as antidifferentiation is the reverse process of differentiation.
Why must the constant of integration, C, be included when finding an indefinite integral?
The constant C is included because the derivative of any constant is zero, meaning a whole family of functions (differing by a constant) are all valid antiderivatives of f(x).
What is the key difference between an antiderivative and an indefinite integral?
An antiderivative, F(x), is a single function whose derivative is f(x), while the indefinite integral, ∫ f(x) dx, represents the entire family of these functions, F(x) + C.