= Antiderivative
{title2=$F'=g$}
{wiki}
= Indefinite integral
{synonym}
An antiderivative recovers a <function> from its local rate of change. For a <real-valued function> $g$ on an <open subset> $U$ of the <real line>, a <differentiable function> $F$ is an antiderivative when
$$
\boxed{F'(x)=g(x)\qquad(x\in U).}
$$
For example, $g(x)=x^2$ has the antiderivatives $F_C(x)=x^3/3+C$ for every real <additive constant> $C$. Their <derivatives> agree, while their values can differ. On an <interval>, the <constant of integration> describes the entire freedom in the answer. https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives[OpenStax Calculus 1, §4.10: antiderivatives and their families]{ref}
\Image[/antiderivative-family-and-initial-value.png]
{title=A family of antiderivatives has one common derivative, and an initial value selects one member}
The indefinite-integral notation $\int g(x)\,dx=F(x)+C$ records this family on an <interval>. For the illustrated family, the extra condition $F(1)=2$ selects $C=5/3$. Thus reversing <differentiation> requires both a rate and a reference value if a unique <function> is wanted.
The <fundamental theorem of calculus> supplies a construction when $g$ is <continuous> on an <interval> $I$. Choose $x_0\in I$ and set $F(x)=\int_{x_0}^x g(t)\,dt$; then $F'=g$. <Integrability> alone does not guarantee a classical antiderivative. The <Heaviside step function> is <Riemann integrable> on bounded <intervals>, but cannot be a <derivative> throughout an <interval> containing its jump: <Darboux theorem for derivatives> requires every intermediate value. https://www.jirka.org/ra/html/sec_mvt.html[Lebl, Basic Analysis I, §4.2.5: the intermediate-value property of derivatives]{ref}
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