Definite vs Indefinite Integral

Both definite and indefinite integrals use the same integration techniques (substitution, by-parts, partial fractions), but they answer fundamentally different questions and produce fundamentally different things.

What each one is

Indefinite integral $\int f(x) \, dx$ — produces a function, the family of antiderivatives:

$\int f(x) \, dx = F(x) + C$

where $F'(x) = f(x)$. The "+C" reminds you that there are infinitely many antiderivatives (any vertical shift works).

Definite integral $\int_a^b f(x) \, dx$ — produces a number, the signed area between the curve $y = f(x)$ and the x-axis on the interval $[a, b]$:

$\int_a^b f(x) \, dx = F(b) - F(a)$

(Fundamental Theorem of Calculus.)

Key differences at a glance

Aspect	Indefinite	Definite
Output	Function $F(x) + C$	Number
Limits	None	$a$ (lower) and $b$ (upper)
"+C" needed	Yes	No (cancels in subtraction)
Geometric meaning	Antiderivative family	Signed area

Worked example

Evaluate both for $f(x) = 2x$.

Indefinite: $\int 2x \, dx = x^2 + C$.

Definite from 0 to 3: $\int_0^3 2x \, dx = [x^2]_0^3 = 9 - 0 = 9$.

The number 9 is the area of the triangle bounded by $y = 2x$, $x = 0$, $x = 3$ — and indeed that triangle has base 3 and height 6, so area $= \frac{1}{2}(3)(6) = 9$. ✓

"Signed" area — what does that mean?

When $f(x) < 0$ on $[a, b]$, the definite integral is negative. It still represents area (in absolute value), but with a sign indicating the curve is below the axis.

Example: $\int_0^\pi \sin x \, dx = 2$ (above axis, positive). $\int_\pi^{2\pi} \sin x \, dx = -2$ (below axis, negative). $\int_0^{2\pi} \sin x \, dx = 0$ (cancels).

If you want the unsigned area, integrate $|f(x)|$ — break at zero crossings.

How they connect: the Fundamental Theorem

The bridge between them is the Fundamental Theorem of Calculus, which says:

1. Differentiation and integration are inverse operations.
2. Definite integrals can be computed by finding any antiderivative (any indefinite integral) and evaluating at the endpoints.

This is why mastering indefinite integrals is the prerequisite for computing definite integrals.

Common mistakes

• Forgetting "+C" on indefinite integrals — half a point off most homeworks.
• Including "+C" on definite integrals — it cancels in $F(b) - F(a)$ and adding it shows confusion.
• Substituting limits before integrating when using u-substitution with definite integrals — change limits to the new variable, or substitute back to $x$ first. Either works, but mixing them causes errors.

Try both with our solver

Drop any integral into the Integral Calculator — toggle between definite (with limits) and indefinite. The AI shows step-by-step techniques and the geometric interpretation.