11.1.1. Antiderivatives

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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 $F(x)+C$.

11.1.1. Antiderivatives

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