Basic Property of the Antiderivative
What is an Antiderivative?
An antiderivative (or primitive function) of a function $f(x)$ is a function $F(x)$ whose derivative equals $f(x)$.
For example, $F(x) = x^3$ is an antiderivative of $f(x) = 3x^2$ because $\frac{d}{dx}(x^3) = 3x^2$.
But wait — is $x^3$ the only antiderivative of $3x^2$? What about $x^3 + 5$? Or $x^3 - 17$?
$$\frac{d}{dx}(x^3 + 5) = 3x^2 \quad \checkmark$$
$$\frac{d}{dx}(x^3 - 17) = 3x^2 \quad \checkmark$$
They all work! This observation leads us to the most fundamental property of antiderivatives.
⭐ The Basic Property of the Antiderivative
Theorem (Basic Property)
If $F(x)$ is an antiderivative of $f(x)$ on an interval $I$, then the most general antiderivative of $f(x)$ on $I$ is:
$$F(x) + C$$
where $C$ is an arbitrary constant (called the constant of integration).
In other words: Any two antiderivatives of the same function differ by a constant.
If $F(x)$ and $G(x)$ are both antiderivatives of $f(x)$, then:
$$G(x) - F(x) = C \quad \text{for some constant } C$$