Table of Contents
An antiderivative is a function whose derivative is a given function. In the context of indefinite integrals, antiderivatives are the objects we are trying to find.
More precisely, given a function $f$, a function $F$ is called an antiderivative (or primitive) of $f$ on an interval if
$$
F'(x) = f(x) \quad \text{for all } x \text{ in the interval.}
$$
If $F$ is one antiderivative of $f$, then every other antiderivative of $f$ on the same interval has the form
$$
F(x) + C,
$$
where $C$ is a constant. This is why indefinite integrals include a constant of integration.
Using the integral notation, the family of all antiderivatives of $f$ is written as
$$
\int f(x)\,dx = F(x) + C.
$$
Here:
- $f(x)$ is called the integrand.
- $dx$ indicates the variable of integration.
- $F(x)$ is any one antiderivative of $f(x)$.
- $C$ is an arbitrary constant.
Because differentiation “loses” constant terms (the derivative of any constant is $0$), antiderivatives are determined only up to an added constant.
In practice, “finding an antiderivative” of $f(x)$ means finding at least one function $F(x)$ such that $F'(x)=f(x)$, and then writing the general form $F(x)+C$.