Definition of a primitive

What is a primitive?

Let f be a function defined on an interval I. We say that F is a primitive of f on I if F is differentiable on I and if, for every x in I:

F'(x) = f(x)

In other words, differentiating F yields f. This is the inverse operation of differentiation.

A simple example

If f(x) = 2x, then F(x) = x² is an antiderivative of f, since (x²)' = 2x.

But G(x) = x² + 3 is also an antiderivative, since (x² + 3)' = 2x. And so is H(x) = x² - 7.

An infinite number of antiderivatives

If F is an antiderivative of f on I, then all functions of the form F(x) + C, where C is any real constant, are also antiderivatives of f on I. We often write: the antiderivatives of f are F(x) + C, where C belongs to R.

Conversely, two antiderivatives of the same function on an interval differ by a constant.

Existence

Theorem (accepted): every continuous function on an interval I has antiderivatives on I.

Common pitfall

Never forget the ‘+C’ when giving the general antiderivative: x³ is not THE antiderivative of 3x², it is ONE antiderivative amongst an infinite number. Also, be careful not to confuse the direction: differentiating f gives f’; finding an antiderivative of f amounts to going back to a function of which f is the derivative.