Recall that the definition of an integral requires the function *f*(*x*) to be bounded on the bounded
interval [*a*,*b*] (where *a* and *b* are two real numbers). It is
natural then to wonder what happens to this definition if

**1**- the function
*f*(*x*) becomes unbounded (we call this case**Type I**); **2**- the interval [
*a*,*b*] becomes unbounded (that is or )(we call this case**Type II**).

**Case Type I:** Consider the function *f*(*x*) defined
on the interval [*a*,*b*] (where *a* and *b* are real numbers). We have
two cases *f*(*x*) becomes unbounded around *a* or unbounded around *b* (see the images below)

and

For the sake of illustration, we considered a positive function. The
integral represents the area of the
region bounded by the graph of *f*(*x*), the x-axis and the lines *x*=*a*
and *x*=*b*. Assume *f*(*x*) is unbounded at *a*. Then the trick behind
evaluating the area is to compute the area of the region bounded by
the graph of *f*(*x*), the x-axis and the lines *x*=*c* and *x*=*b*. Then
we let *c* get closer and closer to *a* (check the figure below)

Hence we have

Note that the integral is well
defined. In other words, it is not an improper integral.

If the function is unbounded at *b*, then we will have

**Remark.** What happened if the function *f*(*x*) is unbounded at
more than one point on the interval [*a*,*b*]?? Very easy, first you
need to study *f*(*x*) on [*a*,*b*] and find out where the function is
unbounded. Let us say that *f*(*x*) is unbounded at and for
example, with . Then you must choose a
number between and (that is )
and then write

Then you must evaluate every single integral to obtain the integral
. Note that the single integrals do
not present a bad behavior other than at the end points (and not for
both of them).

**Example.** Consider the function defined on [0,1]. It is easy to see that *f*(*x*)
is unbounded at *x* = 0 and .
Therefore, in order to study the integral

we will write

and then study every single integral alone.

**Case Type II:** Consider the function *f*(*x*) defined
on the interval or . In other words, the
domain is unbounded not the function (see the figures below).

and

The same as for the Type I, we considered a positive function just for the sake of illustrating what we are doing. The following picture gives a clear idea about what we will do (using the area approach)

So we have

and

**Example.** Consider the function defined on . We have

On the other hand, we have

Hence we have

It may happen that the function *f*(*x*) may have Type I and Type II
behaviors at the same time. For example, the integral

is one of them. As we did before, we must always split the integral into a sum of integrals with one improper behavior (whether Type I or Type II) at the end points. So for example, we have

The number 1 may be replaced by any number between 0 and since the function has a Type I behavior at 0 only and of course a Type II behavior at .

**Convergence** and **Divergence** of improper integrals will be discussed in the next pages.

**
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Tue Dec 3 17:39:00 MST 1996

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