Antiderivative
An antiderivative of a function f is any function F whose derivative equals f, representing the reverse process of differentiation.
Formula
F'(x) = f(x)
Definition
An antiderivative is the reverse of a derivative: if the derivative of $F$ is $f$, then $F$ is an antiderivative of $f$, "undoing" differentiation. Formally, $F$ is an antiderivative of $f$ if $F'(x) = f(x)$; antiderivatives are not unique, since if $F$ is one, then $F + C$ for any constant $C$ is another. If $f$ is continuous on $[a, b]$, then $F(x) = \int_a^x f(t)\, dt$ is the unique antiderivative of $f$ with $F(a) = 0$, the content of the First Fundamental Theorem of Calculus; but an antiderivative is not always expressible in closed form, for instance $\int e^{-x^2}\, dx$.
Example
The derivative of $x^3$ is $3x^2$, so $x^3$ is an antiderivative of $3x^2$. Antiderivatives of $6x^2$ include $F(x) = 2x^3$, or $2x^3 + 5$, or $2x^3 - 17$; all share the same derivative, so the general form is $2x^3 + C$. The function $e^{-x^2}$ has no elementary antiderivative; its integral defines the error function $\text{erf}(x) = (2/\sqrt{\pi}) \int_0^x e^{-t^2}\, dt$, used throughout statistics and physics.
Key Insight
Finding an antiderivative is like asking: "What function, when differentiated, gives me this one?" The constant $C$ represents the "lost information" of differentiation, since two functions differing only by a constant have identical derivatives. Risch's algorithm (1969) decides whether an elementary antiderivative exists and finds it if so, placing the search for antiderivatives on a rigorous algorithmic foundation.