`int_e^(e^4) (dx)/(xsqrt(ln(x)))` Evaluate the definite integral.

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.`