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**).

**improper**.

**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)

*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)

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

**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 .