AP Calculus AB and BC

Definite vs Indefinite Integral

A definite integral has bounds and evaluates to a number. An indefinite integral has no bounds and evaluates to a family of functions plus a constant of integration. One is a value, the other is a function, and that single difference explains when the constant is needed.

Definite integral

Use when: You want an accumulated amount, an area, or a net change between two specific bounds.

Indefinite integral

Use when: You want the antiderivative itself, usually to solve a differential equation or to use later.

Side by side

	Definite	Indefinite
Notation	$\int_a^b f(x)\,dx$	$\int f(x)\,dx$
Result	A number	A family of functions
Needs $+ C$	No	Yes
Depends on the variable	No, it is a constant	Yes

The constant of integration disappears from a definite integral because it cancels in the subtraction: $\left(F(b) + C\right) - \left(F(a) + C\right)$ leaves $F(b) - F(a)$ . That is why any antiderivative works for evaluating one.

Because a definite integral is a number, the variable of integration is only a placeholder. The expressions $\int_a^b f(x)\,dx$ and $\int_a^b f(t)\,dt$ are the same number, which is not true of the indefinite versions.

The mistake

Leaving off the constant on an indefinite integral. Without it the answer names one antiderivative rather than all of them, and it is a routine deduction.

Frequently asked questions

Can a definite integral be negative?

Yes. It measures net signed accumulation, so region below the axis counts negatively. For true area, integrate the absolute value.

In the CED: Unit 6: Integration and Accumulation