Antiderivative
Calculus & Advanced MathAn 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. It "un-does" differentiation.
Example
The derivative of $x^3$ is $3x^2$. So $x^3$ is an antiderivative of $3x^2$. Working backwards is the key idea.
Key Insight
Finding an antiderivative is like asking: "What function, when differentiated, gives me this one?"
Definition
$F$ is an antiderivative of $f$ if $F'(x) = f(x)$. Antiderivatives are not unique: if $F$ is one, then $F + C$ for any constant $C$ is another. The full family is written $F(x) + C$.
Example
Antiderivatives of $6x^2$: $F(x) = 2x^3$, or $2x^3 + 5$, or $2x^3 - 17$. All share the same derivative. The general form is $2x^3 + C$.
Key Insight
The constant $C$ represents the "lost information" of differentiation: two functions differing only by a constant have identical derivatives.
Definition
If $f$ is continuous on $[a, b]$, then $F(x) = \int_a^x f(t)\, dt$ is the unique antiderivative of $f$ with $F(a) = 0$. This is the content of the First Fundamental Theorem of Calculus. The antiderivative is not always expressible in closed form (e.g., $\int e^{-x^2}\, dx$).
Example
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
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.