# What is the integral of #x/(x-2)# from 0 to 3?

##### 1 Answer

Depending on where you are in your learning of calculus, the best answer may be "Is not defined" or the best answer may be "The integral diverges".

The definite integral over interval

That is: we start with a function

(Some treatments initially require the stronger condition: "Let

Therefore, using the intial definition (before you learn about "Improper Inetgrals") the best answer is this integral is not defined.

**Improper integrals** extend the initial definition by allowing the endpoints

outside the domain of

or for the interval to lack left and/or right endpoints (infinite intervals).

**General Method**

Replace the "problem number" (or

If the limit exists, then we say that the integral converges. If the limit fail to exist (possibly by "being" infinite), then then integral diverges.

**Example 1**

To try to evaluate

(Eric gives and excellent description of finding the indefinite integral, so I won't go through that again, though I'll write

But this limit does not exist. (

So the integral diverges.

Because one of the two integrals needed diverges, there is no need to check the other.

**That answers the question this is posted under.**

**Example 2**

It will probably be helpful to many students to see an example of an improper integral that converges.

When we extend the definition to Improper integrals (I always want to say " *so-called* Improper Integrals")

We try to evaluate:

The integral converges to

(Opinions vary on whether we should say "the integral **equals**

If you have additional questions of would like to see additional examples, post new questions or send me a note and I'll suggest a question.