AP Calculus BC 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.
How are the processes of determining derivatives and antiderivatives related?
They are inverse processes. To find an antiderivative of a function, one must find a function whose derivative is the original function.
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How are the processes of determining derivatives and antiderivatives related?
They are inverse processes. To find an antiderivative of a function, one must find a function whose derivative is the original function.
If you know the derivative of $\sin(x)$ is $\cos(x)$, how do you determine an antiderivative of $\cos(x)$?
By using knowledge of derivatives in reverse, an antiderivative of $\cos(x)$ is $\sin(x)$. The indefinite integral is $\int \cos(x) dx = \sin(x) + C$.
What is the relationship between an antiderivative $F(x)$ and the original function $f(x)$?
The relationship is that the derivative of the antiderivative equals the original function, expressed as $F'(x) = f(x)$.
Can an antiderivative be found for every function in a simple, standard form?
No, it is not always possible. Many functions do not have closed-form antiderivatives.
Why is $F(x) = x^2 + 10$ an antiderivative of $f(x) = 2x$?
It is an antiderivative because its derivative, $F'(x)$, is equal to the original function $f(x)=2x$.
What does the notation $\int f(x) dx$ represent?
It represents the indefinite integral of the function f, which is the family of all its antiderivatives.
In the expression $\int f(x) dx = F(x) + C$, what is the term '$+ C$' called and what does it represent?
The term '$+ C$' is the constant of integration, representing any constant value, because the derivative of a constant is zero.
What is the key prerequisite knowledge needed to determine antiderivatives of functions?
The key prerequisite is a strong knowledge of derivatives, as finding antiderivatives is the reverse process.
What provides the foundation for the rules of finding antiderivatives?
Differentiation rules provide the foundation for finding antiderivatives, as antidifferentiation is the reverse process of differentiation.
Define an indefinite integral.
An indefinite integral, written $\int f(x) dx$, can be expressed as $F(x) + C$, where $F'(x) = f(x)$ and $C$ is any constant.