Given the derivative of a function, find the original function.

For example, suppose we are given the following derivatives.

f ′ ( x ) = 2 , g ′ ( x ) = 3 x 2 , s ′ ( t ) = 4 t

Our problem is to determine the functions f, g and s that have these derivatives. If we make some educated guesses, we might come up with the following functions.

f ( x ) = 2 x	because	d d x [ 2 x ] = 2
g ( x ) = x 3	because	d d x [ x 3 ] = 3 x 2
s ( t ) = 2 t 2	because	d d t [ 2 t 2 ] = 4 t

This operation of determining the original function from its derivative is the inverse operation of differentiation, and we call it

antidifferentiation.

Definition of an Antiderivative

A function F is called an antiderivative of a function   f   if for every x in the domain of   f

F ' (x) = f (x)

REMARK
We will use the phrase " F(x) is an antiderivative of f(x) " synonymously with " F is an antiderivative of f ".

If F(x) is an antiderivative of f(x), then F(x) +C, where C is any constant, is also an antiderivative of f(x). For example,

F ( x ) = x 3

G ( x ) = x 3 - 5

and

H ( x ) = x 3 + 0.3

are antiderivatives of 3 x 2

d d x [ x 3 ] = d d x [ x 3 - 5 ] = d d x [ x 3 + 0.3 ] = 3 x 2

As it turns out, all the antiderivatives of 3 x 2 are of the form x 3 + C . The point is that the process of antidifferentiation does not determine a unique function but rather it determines a family of functions, each differing from others by a constant.


Notation for Antiderivatives and Indefinite Integrals

The antidifferentiation process is also called integration and is denoted by the symbol ∫ , called an integral sign. The symbol

∫ f ( x ) d x

is called the indefinite integral of f(x), and it denotes the family of antiderivatives of f(x). That is, if F ' (x) = f (x) for all x, then

∫ f ( x ) d x = F ( x ) + C

where f(x) is called the integrand and C, the constant of integration.

The differential dx in the indefinite integral identifies the variable of integration. That is, the symbol ∫ f ( x ) d x denotes the " antiderivative of f with respect to x " just as the symbol dy / dx denotes the " derivative of y with respect to x ".

Definition of Integral Notation for Antiderivatives

The notation

∫ f ( x ) d x = F ( x ) + C

where C is an arbitrary constant, means that F is an antideri-vative of f. That is, F ' (x) = f (x) for all x in the domain of f.

EXAMPLE
Using the integral notation, we can rewrite the three antiderivatives given at the beginning of this section as follows.

∫ 2 d x = 2 x + C

∫ 3 x 2 d x = x 3 + C

∫ 4 t d t = 2 t 2 + C

At this moment you do not know the rules for finding the antiderivatives of some functions. You can test only your ideas about their work.