find
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.
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.
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 $\displaystyle x=0\text{ or }x=1$.