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

##### 3 Answers

There are no differences, the two words are synonymous.

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

Some (for instance, textbook author James Stewart) make a distinction. What Stewart refers to as "the most general" antiderivative of

The indefinite integral of

So

**Particular Antiderivatives**

A particular antiderivative of

For example:

is a particular antidervative of

And:

is a different particular antidervative of

**Definite integrals**

The definite integral of

For example:

(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.)

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 **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 **define** an antiderivative of

In the case where