# What is the difference between an antiderivative and an integral?

Mar 17, 2015

There are no differences, the two words are synonymous.

Mar 19, 2015

It depends on a couple of things. Which antiderivative, the general or a particular? which integral definite or indefinite? And, who are we asking?

General Antiderivative and Indefinite Integral:

Many mathematicians do not distinguish the indefinite integral and the general antiderivative. In either case for function $f$ the "answer" is $F \left(x\right) + C$ where $F ' \left(x\right) = f \left(x\right)$..

Some (for instance, textbook author James Stewart) make a distinction. What Stewart refers to as "the most general" antiderivative of $f$, admits different constants at each discontiuity of $f$. For example, he would answer that the most general antiderivative of $\frac{1}{x} ^ 2$ is a piecewise defined function:

$F \left(x\right) = \frac{- 1}{x} + {C}_{1}$ for $x < 0$ and $\frac{- 1}{x} + {C}_{2}$ for $x > 0$.

The indefinite integral of $f$, in this treatment, is always an antiderivative on some interval on which $f$ is continuous.

So $\int \frac{1}{x} ^ 2 \mathrm{dx} = - \frac{1}{x} + C$, where it is understood that the domain is restricted to some subset of either the positive reals or a subset of the negative reals.

Particular Antiderivatives
A particular antiderivative of $f$ is a function $F$ (rather than a family of functions) for which $F ' \left(x\right) = f \left(x\right)$.

For example:
$F \left(x\right) = \frac{- 1}{x} + 5$ for $x < 0$ and $\frac{- 1}{x} + 1$ for $x > 0$.
is a particular antidervative of $f \left(x\right) = \frac{1}{x} ^ 2$

And:
$G \left(x\right) = \frac{- 1}{x} - 3$ for $x < 0$ and $\frac{- 1}{x} + 6$ for $x > 0$.
is a different particular antidervative of $f \left(x\right) = \frac{1}{x} ^ 2$.

Definite integrals

The definite integral of $f$ from $a$ to $b$ is not a function. It is a number.
For example:
${\int}_{1}^{3} \frac{1}{x} ^ 2 \mathrm{dx} = \frac{2}{3}$.

(To further complicate matters, this definite integral can be found, using the Fundamental Theorem of Calculus, Part 2, by finding the/an indefinite integral / general antiderivative first, then doing somearithmetic.)

Mar 19, 2015

Your question is related to what was truly the "key insight" in the development of calculus by Isaac Newton and Gottfried Leibniz.

Focusing on functions that are never negative, this insight can be phrased as: "Antiderivatives can be used to find areas (integrals) and areas (integrals) can be used to define antiderivatives". This is the essence of the Fundamental Theorem of Calculus.

Without worrying about Riemann sums (after all, Bernhard Riemann lived almost 200 years after Newton and Leibniz anyway) and taking the notion of area as an intuitive (undefined) concept, for a continuous non-negative function $f \left(x\right) \setminus \ge q 0$ for all $x$ with $a \setminus \le q x \setminus \le q b$, just think of the definite integral symbol $\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ as representing the area under the graph of $f$ and above the $x$-axis between $x = a$ and $x = b$. If another function $F$ can be found so that $F ' \left(x\right) = f \left(x\right)$ for all $a \setminus \le q x \setminus \le q b$, then $F$ is called an antiderivative of $f$ over the interval $\left[a , b\right]$ and the difference $F \left(b\right) - F \left(a\right)$ equals the value of the definite integral. That is, $\setminus {\int}_{a}^{b} f \left(x\right) \mathrm{dx} = F \left(b\right) - F \left(a\right)$. This fact is useful for finding the value of a definite integral (area) when a formula for an antiderivative can be found.

Conversely, if we make the upper limit of the integral symbol a variable, call it $t$, and define a function $F$ by the formula $F \left(t\right) = \setminus {\int}_{a}^{t} f \left(x\right) \mathrm{dx}$ (so $F \left(t\right)$ is really the area under the graph of $f$ between $x = a$ and $x = t$, assuming $a \setminus \le q t \setminus \le q b$), then this new function $F$ is well-defined, differentiable, and $F ' \left(t\right) = f \left(t\right)$ for all numbers $t$ between $a$ and $b$. We have used an integral to define an antiderivative of $f$. This fact is useful for approximating values of an antiderivative when no formula for it can be found (using numerical integration methods like Simpson's rule). For instance, it's used all the time by statisticians when approximating areas under the Normal curve. The values of a special antiderivative of the standard Normal curve are often given in a table in statistics books.

In the case where $f$ has negative values, the definite integral must be thought of in terms of "signed areas".