Answer
$\frac{1}{3}{\left( {\ln x} \right)^3} + C$
Work Step by Step
$$\eqalign{
&\text{Let }I= \int {\frac{{{{\left( {\ln x} \right)}^2}}}{x}} dx \cr
& {\text{Let }}u = \ln x,{\text{ then }}du = \frac{1}{x}dx \cr
& {\text{Applying the substitution}}{\text{, we obtain}} \cr
& I = \int {{u^2}} du \cr
& {\text{Integrate apply the power rule }}\int {{u^n}du} = \frac{{{u^{n + 1}}}}{{n + 1}}{\text{ + C }} \cr
& I = \frac{1}{3}{u^3} + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}u = \ln x \cr
& I= \frac{1}{3}{\left( {\ln x} \right)^3} + C \cr} $$
