need to find the definite integral of lnx/x on [e, infinity]?

Question

thanks for the help

Answer 1

Let u = ln x, du = 1/x dx.
∫(e to b) (ln x / x) dx
= ∫(1 to b) (u du)
= (b^2 - 1)/2
As b -> ∞, so the integral goes to ∞. So the (infinite) integral is undefined.

Answer 2

∞
∫( ln(x)/x dx )
e

Since this is an improper integral, you have to express it as a limit.

lim ∫ (e to t, ln(x)/x dx )
t → ∞

lim ∫ (e to t, ln(x) (1/x) dx )
t → ∞


To evaluate the integral, we need to use substitution.
Let u = ln(x). {When x = e, u = ln(e) = 1. When x = t, u = ln(t) }
du = (1/x) dx

After the substitution, our bounds change, and the integral becomes

lim ∫ (1 to ln(t), u du )
t → ∞

Which is easily evaluated.

lim [ (1/2)u² ] { evaluated from 1 to ln(t) }
t → ∞

lim [ (1/2)[ln(t)]² - (1/2)[ln(1)]² ]
t → ∞

lim [ (1/2)[ln(t)]² - (1/2)[0]² ]
t → ∞

lim (1/2)[ln(t)]²
t → ∞

But as t approaches infinity, ln(t) approaches infinity, so the limit does not exist and the integral diverges.