Additive constant
= Additive constant
{title2=$F(x)+C$}
A constant added to a <function> changes its reference level without changing its <derivative>:
$$
\frac{d}{dx}\bigl(F(x)+C\bigr)=F'(x).
$$
Geometrically, it is a vertical <translation> of the graph. For $F(x)=x^3/3$, changing $C$ moves every plotted value by the same amount while preserving every tangent slope. This is the geometric reason that <antiderivatives> form a family.
When $F$ represents a <physical quantity>, the <physical dimension> of $C$ must match that of $F$. When <position> is obtained from <velocity>, $C$ is a position offset. An unknown starting position remains unknown even when the rate of change is known exactly.
When describing the family of <antiderivatives> on an <interval>, the arbitrary additive constant is the <constant of integration>. On a disconnected open domain, the corresponding freedom is a <locally constant function>, as in <antiderivatives on disconnected domains>, rather than necessarily one constant on the whole domain.