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Consider an integrable function
and two numbers *c* and *x* in [*a,b*].
Then the integral exists. If the number
*c* is fixed, the integral depends on *x*.

If, for example, *f(x)=x* the integral can be
calculated by a simple geometrical method. You must only observe
that the integral can be obtained by calculating areas of
trapeziums or triangles, with the appropriate sign.

You always obtain: . The result is a new
function, that we call *an integral function* of
*f*, and represent by *F(x)*: .

Observe, in the previous integral, that the
variable *x* is used in two different places:

- as the variable of the integrand function;
- as the upper bound of the integral.

In order to avoid confusion, usually one writes the same integral using a different letter for the variable of the integrand function: ; in fact the variable of the integrand function is not important, and any letter works.

This is a new way to construct functions. In some cases, as the previous example shows, the new function can be obtained by elementary methods, but we'll see that in many other cases the new function is not an elementary one, so this is a completely new technique to construct functions.

Obviously you could have considered the integral bounds in
reverse order, but the function you obtain is simple the
opposite one: *-F(x)*.

You can also consider the composition technique to obtain more functions; for example, slightly modifying the previous integral you have: . A still more complex situation can be obtained, for example, as follows: .

The function *f* must be integrable between
*c* and *x* (or *x* and *c*,
if * x* is less than *c*). For example
the function , exists only for
*x>*0, while the function
exists only for *x*<0.

Observe, in the example at the beginning of this page, that the
derivative of *F(x)* is *f(x)*, and the lower
bound *c* of the integral is not important for this. We
may write: , whatever is the point
*c*, provided the integral exists.

We give the following

Given an integrable function
and two numbers *c* and *x* in
[*a,b*], is the
** integral function** of

This function plays a critical role in the theory of Riemann integrals.

copyright 2000 et seq. maddalena falanga & luciano battaia

first published on january 07 2003 - last updated on september 01
2003