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

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Answer 1 · 2016-10-29T20:40:58

You can't as the integral is divergent

Answer 2 · 2016-10-29T20:55:33

The integral is not convergent.

$d/dx(log_ex)^2 = 2/xlog_ex$ 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.

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