Integral of 1/(x*ln(x)) converging or diverging?

[ndnbolla]

In order to find the integral of 1/(x*ln(x)) dx, I tried using the substitution method where

u = 1/x and dv = 1/ln(x) dx .

Then du = ln(x) dx.

However this is where I got stuck.

What would v equal? Or is their another way I should be approaching this integral in order to find if it is diverging or converging and if converging, to what value?

 

[siddharth]

First of all, is it a definite integral? What are the limits of integration?

Then, if u=1/x then du is not ln(x) dx. Why don't you try the substitution u=ln(x) and see what happens?

 

[ndnbolla]

My mistake and sorry for not mentioning the limits. Its an improper integral with lower limit 2 and upper limit infinity.

I think I found the solution...

u=ln(x)
du=(1/x)

int(1/u) du

And then solve for the integral, with the lower limit being ln(2) and the upper limit being infinity.

It ends up diverging, correct?

 

[siddharth]

Yes, I think you're right.