Constant of integration

The $+C$ appended to an indefinite integral, capturing every antiderivative at once.

The constant of integration, written $+C$ , is appended to every indefinite integral because differentiation destroys constants: $\frac{d}{dx}[F(x)+C]=F'(x)$ for ANY value of $C$ . Dropping the $+C$ silently claims a single antiderivative is the only one, which is false — an initial condition (a known point on the graph) is what pins $C$ down to a specific value.

Examples

$\int 6x\,dx = 3x^2+C$ — every value of $C$ gives a valid antiderivative of $6x$ .
Given $F(0)=5$ and $F(x)=x^2+C$ , solving $0^2+C=5$ pins down $C=5$ .