Indefinite Integral
The indefinite integral of a function is the family of all its antiderivatives, written with a constant C to represent all possibilities.
Formula
\int f(x)\, dx = F(x) + C
Definition
The indefinite integral finds all antiderivatives of a function at once; the answer always includes "$+ C$" to represent the unknown constant that differentiation erases. Written $\int f(x)\, dx = F(x) + C$, it represents the general antiderivative family, and key rules mirror differentiation: $\int x^n\, dx = x^{n+1}/(n+1) + C$ (for $n \neq -1$), $\int e^x\, dx = e^x + C$, $\int \cos(x)\, dx = \sin(x) + C$. More abstractly, all antiderivatives of $f$ form a coset $F + \ker(d/dx)$, where $\ker(d/dx)$ is the constant functions on a connected domain; on disconnected domains, the "constant" can differ on each component.
Example
$\int 3x^2\, dx = x^3 + C$, verified by differentiating $x^3 + C$ to get back $3x^2$. Term by term, $\int (4x^3 + 2x - 5)\, dx = x^4 + x^2 - 5x + C$, using the power rule in reverse. On a disconnected domain, $\int 1/x\, dx = \ln|x| + C$ can carry different constants $C_1$ and $C_2$ on $(-\infty, 0)$ and $(0, \infty)$ separately.
Key Insight
The "$+ C$" is not laziness, it represents an infinity of valid answers, all differing only by a vertical shift, and every differentiation rule has a corresponding integration rule, so learning them in pairs makes both easier to remember. The indefinite integral is not a single function but an equivalence class of functions, a distinction that matters in distributional calculus and in the theory of differential equations.