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
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)
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).
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
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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Mohamed A. Khamsi