AP Calculus AB and BC

Antiderivative vs Indefinite Integral

An antiderivative is any single function whose derivative is f. The indefinite integral is the entire family of them, which is why it carries a plus C. So x squared over 2 is an antiderivative of x, while the indefinite integral of x is x squared over 2 plus C.

An antiderivative

Use when: You need one specific function, such as the F in the Fundamental Theorem where the constant cancels anyway.

The indefinite integral

Use when: You are asked to integrate with no limits, where the complete answer is the whole family.

Side by side

	An antiderivative	The indefinite integral
How many functions	One	Infinitely many
Carries $+C$	No	Yes, always
Notation	$F(x)$ with $F' = f$	$\int f(x)\,dx$
Example for $f(x) = x$	$\frac{x^{2}}{2}$ , or $\frac{x^{2}}{2} + 7$	$\frac{x^{2}}{2} + C$
Used in FTC as	Any single $F$	Not directly

The reason the family is exactly "one antiderivative plus a constant" is a corollary of the Mean Value Theorem: two functions with the same derivative on an interval differ by a constant. Without that theorem, $+C$ would be a guess rather than a complete answer.

Why the FTC lets you drop the C

In a definite integral you subtract F(b) minus F(a), and any constant appears in both terms and cancels. That is why the Fundamental Theorem says ANY antiderivative works.

Frequently asked questions

What is the difference between an antiderivative and an indefinite integral?

An antiderivative is one function; the indefinite integral is the whole family, which is what the $+C$ records.

Why do I need +C?

Because infinitely many functions share the same derivative, differing only by a constant. Omitting it gives an incomplete answer.

Does the constant matter in a definite integral?

No. It appears at both endpoints and cancels in the subtraction, which is why the FTC lets you use any antiderivative.

In the CED: Unit 6: Integration and Accumulation