Antiderivatives

An antiderivative is a differentiable function F whose derivative is equal to $f$ (i.e., $F'=f$). The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Let's consider the case of function $F(x) = \frac{x^3}{3}$, which is an antiderivative of $f(x) = x^2$. As the derivative of a constant is zero, $x^2$ will have an infinite number of antiderivatives, such as $\frac{x^3}{3} + 0$, $\frac{x^3}{3} + 7$, $\frac{x^3}{3} - 42$, $\frac{x^3}{3} + 293$, etc. Therefore, all the antiderivatives of $x^2$ can be obtained by adding the value of $C$ in $F(x) = \frac{x^3}{3} + C$, where $C$ is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.

Antiderivatives are important because they can be used to compute definite integrals with the fundamental theorem of calculus: if $F$ is an antiderivative of the integrable function $f$, and $f$ is continuous over the interval $[a, b]$, then

$\displaystyle{\int_{a}^{b}f(x)dx = F(b) - F(a)}$

Because of this rule, each of the infinitely many antiderivatives of a given function $f$ is sometimes called the "general integral" or "indefinite integral" of $f$, and is written using the integral symbol with no bounds:

$\displaystyle{\int f(x)dx}$

If $F$ is an antiderivative of $f$, and the function $f$ is defined on some interval, then every other antiderivative $G$ of $f$ differs from $F$ by a constant: there exists a number $C$ such that $G(x) = F(x) + C$ for all $x$. $C$ is called the arbitrary constant of integration. If the domain of $F$ is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance:

$F (x) =\begin{cases} \ -1/x + C_{1} \text{ if } x<0 \\ \ -1/x + C_{2} \text{ if } x>0 \end{cases}$

is the most general antiderivative of $f(x) = \frac{1}{x^2}$ on its natural domain of:

$(-\infty ; 0)\bigcup (0; \infty )$.