Indefinite Integral
Calculus & Advanced MathThe 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.
Example
$\int 3x^2\, dx = x^3 + C$. You can verify by differentiating $x^3 + C$: you get $3x^2$, which matches the original function.
Key Insight
The "$+ C$" is not laziness. It represents an infinity of valid answers, all differing only by a vertical shift.
Definition
The indefinite integral $\int f(x)\, dx = F(x) + C$ represents the general antiderivative family. Key rules mirror differentiation: $\int x^n\, dx = x^{n+1}/(n+1) + C$ (for $n \neq -1$), $\int e^x\, dx = e^x + C$, $\int \cos(x)\, dx = \sin(x) + C$.
Example
$\int (4x^3 + 2x - 5)\, dx = x^4 + x^2 - 5x + C$, using the power rule in reverse on each term.
Key Insight
Every differentiation rule has a corresponding integration rule. Learning them in pairs (power rule for derivatives and integrals together) makes both easier to remember.
Definition
The indefinite integral is a coset in the function space: all antiderivatives of $f$ form the coset $F + \ker(d/dx)$, where $\ker(d/dx) = \{\text{constant functions}\}$ on a connected domain. On disconnected domains, the "constant" can differ on each component.
Example
$\int 1/x\, dx = \ln|x| + C$ on each connected component of $\mathbb{R} \setminus \{0\}$. On $(-\infty, 0)$ and $(0, \infty)$ separately, the constants of integration can be different values $C_1$ and $C_2$.
Key Insight
The indefinite integral is not a function but an equivalence class of functions. This distinction matters in distributional calculus and in the theory of differential equations.