Calculus: Early Transcendentals, Chapter 5, 5.5, Section 5.5, Problem 69

You need to use the following substitution ln x=u, such that:
ln x=u=>(dx)/x= du
int_e^(e^4) (dx)/(x*sqrt(ln x)) = int_(u_1)^(u_2) (du)/(sqrt u)
int_(u_1)^(u_2) (du)/(sqrt u) = 2sqrt u|_(u_1)^(u_2)
Replacing back ln x for u yields:
int_e^(e^4) (dx)/(x*sqrt(ln x)) = 2sqrt (ln x)|_e^(e^4)
Using Leibniz-Newton theorem yields:
int_e^(e^4) (dx)/(x*sqrt(ln x)) = 2sqrt (ln e^4) - 2sqrt (ln e)
int_e^(e^4) (dx)/(x*sqrt(ln x)) = 2sqrt 4 - 2sqrt 1
int_e^(e^4) (dx)/(x*sqrt(ln x)) = 4 - 2
int_e^(e^4) (dx)/(x*sqrt(ln x)) = 2
Hence, evaluating the definite integral, yields int_e^(e^4) (dx)/(x*sqrt(ln x)) = 2.