Does the Integral of 1/ln(x) from 0 to 1 Diverge or Converge?

[Dell]

sorry that i do not know the correct english terminology, hope you understand

i need to look into tis integral
$$\int$$dx/ln(x) (from 0 to 1)

and say if it comes to a number or to infinity or what, don't necessarily have to gve the answer as a number, can either comapre it to another simpler function or work out the integral.

if F(x)<G(x) and G(x) collects to an actual number than so will F(x), etc
again, hope I am clear and understood

 

[Dick]

Clearly the only place you could have a problem with convergence is near x=1. Find something to compare with in a small interval near x=1, think about the tangent line to the curve y=ln(x) there.

 

[Dell]

i cannot find such a function of x that is always bigger or always smaller than 1/lnx. could you please show me

 

[Dick]

It doesn't have to be always bigger or always smaller. It only has to be bigger or smaller where it counts. Can you find f(x) so that 1/|f(x)|<1/|ln(x)| only on the interval [0.9,1] and 1/|f(x)| diverges over that interval? Make f a linear function. What is the tangent to y=ln(x) at x=1?