1. ## vip integral

find
$\LARGE \int_{0}^{2}\frac{x}{lnx}dx$

2. Originally Posted by dapore
find
$\LARGE \int_{0}^{2}\frac{x}{lnx}dx$
It doesn't converge.

3. Originally Posted by Drexel28
It doesn't converge.
Why?

4. Well, if you use integration by parts, regardless of what you get for the other forms in: $\displaystyle \int u*dv = u*v-\int du*v$

The form of u*v will be undefined for one of your functions.

Which one and why I leave to you, but it should be obvious if you use IBP. Are you that far yet - you'd have to be to do this problem.

5. Originally Posted by dapore
find
$\LARGE \int_{0}^{2}\frac{x}{lnx}dx$
Consider

$\displaystyle f(x)=\frac x{\ln(x)}$

The domain of this function is $\displaystyle d = [0, 1) \cup (1, \infty)$

and

f(x) is approaching $\displaystyle -\infty$ if x approaches 1 from the left;
f(x) is approaching $\displaystyle +\infty$ if x approaches 1 from the right.

That's the reason why your integraL cann't exist.

6. I want more clarification, I should be grateful to you

7. Originally Posted by dapore
I want more clarification, I should be grateful to you
It's easy:

1. Draw the graph of $\displaystyle f(x)=\frac x{\ln(x)}$

2. Draw the borders of the integral and it is obvious that the integral doesn't exist.

8. Originally Posted by dapore
I want more clarification, I should be grateful to you
There is absolutely no need for you to screem at us.
Using size five type is uncalled for.

Now here is a question for you: "Do you understand an improper integral?"
Note the function $\displaystyle \frac{x}{\ln(x)}$ is not defined for either $\displaystyle x=0\text{ or }x=1$.