Question

How do you integrate `ln(x)/(x)dx` from 1 to infinity?

Answer 1

You can't as the integral is divergent

Answer 2

The integral is not convergent.

Explanation:

`d/dx(log_ex)^2 = 2/xlog_ex` then

`int_1^a 1/xlog_ex dx = 1/2((log_e a)^2-(log_e1)^2)= 1/2(log_e a)^2`

but

`log_e a` is a monotonic strict increasing function of `a` so

`lim_(a->oo)int_1^a log_e(x)/xdx=oo` is not convergent.