How do I solve this natural log integration problem?

[efekwulsemmay]

Homework Statement

\int^{4}_{2} \frac{dx}{x\left(lnx\right)^{2}}

Homework Equations

Let u=lnx
du=\frac{1}{x}dx
x=2 \rightarrow u=ln2
x=4 \rightarrow u=ln4

The Attempt at a Solution

so with the u substitution we have:

\int^{ln4}_{ln2} \frac{1}{u^{2}}du

which goes to:

lnu^{2}\right|^{ln4}_{ln2}

then:

2\cdot lnu\right|^{ln4}_{ln2}

and when we work it out we get:

2\cdot\left[ln\left(ln4\right)-ln\left(ln2\right)\right]

and then:

2\cdot ln\left(\frac{ln4}{ln2}\right)

This is where I am stuck. I am supposed to get:

\frac{1}{ln4}

and I have no idea how they got that. Any help would be appreciated.

 

[Samuelb88]

then integral of 1/u^2 evaluates to -1/u.