Answer
$$\frac{1}{2}{\left( {\ln 4} \right)^2}$$
Work Step by Step
$$\eqalign{
& \int_1^4 {\frac{{\ln x}}{x}dx} \cr
& {\text{Let }}u = \ln x,{\text{ }}du = \frac{1}{x}dx \cr
& {\text{The new limits of integration are}} \cr
& x = 4 \to u = \ln 4 \cr
& x = 1 \to u = 0 \cr
& {\text{Substituting}} \cr
& \int_1^4 {\frac{{\ln x}}{x}dx} = \int_0^{\ln 4} {udu} \cr
& {\text{Integrating}} \cr
& = \frac{1}{2}\left[ {{u^2}} \right]_0^{\ln 4} \cr
& = \frac{1}{2}{\left( {\ln 4} \right)^2} \cr} $$
