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

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".

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

General Method

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.

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:

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.

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To solve the integral of ( \frac{x}{x-2} ) from 0 to 3, we can split the integral into two parts using partial fraction decomposition. The integral will be equal to the integral of ( \frac{x-2+2}{x-2} ) from 0 to 3. This simplifies to the integral of ( \left(1 + \frac{2}{x-2}\right) ) from 0 to 3. Integrating each part separately gives ( x + 2\ln|x-2| ). Evaluate this expression at 3 and 0, then subtract the result at 0 from the result at 3. After substitution, the result will be ( (3 + 2\ln|1|) - (0 + 2\ln|2|) ), which simplifies to ( 3 + 2\ln(1) - 2\ln(2) ). Finally, ( \ln(1) = 0 ), so the result is ( 3 - 2\ln(2) ), approximately 0.3069.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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