Question

How do you integrate `ln(x) / x` from 4 to infinity?

Answer 1

the integral does not converge.

Explanation:

for the basic integration, spot the pattern:

`d/dx (ln^alpha x) = alpha ln^(alpha - 1) x * 1/x`

So

`d/dx (color{red}{1/2} ln^2 x) = 1/2 * 2 ln x * 1/x = ln x * 1/x`

So

`int_4^oo \ ln(x) / x \ dx = int_4^oo \ d/dx (1/2 ln^2 x) \ dx`

`= lim_(t to oo) (1/2 ln^2 x)_4^t`

the problem being, the integral does not converge.