Determine weather the improper integral converges or diverges.
Evaluate if convergent.  Please show work.

 inf  ___1___  dx
S     x ln x
 2

Hi again cousinit!!

I will use again a variables change:

Take t = ln x, then dt = dx/x .

And the antiderivative of 1/t is ln(t) (t is positive in this problem).

Now we have:

 +oo                  +oo
S    ___1___  dx =  S     dt/t =
 2    x ln x         ln(2)


                 =  lim   [ln(a) - ln(ln(2))] = +inf
                   a->+oo 

The improper integral diverges.

Regarding the  change of variables, see the following page for references:
"Change of Variables or Substitution":
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/techniques/substitution.html

  
I hope this helps. Please feel free for request for any clarification needed.

Regards.
livioflores-ga