Answer
\[\frac{{\,{{\left( {1 + 3\ln x} \right)}^3}}}{9} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{\,{{\left( {1 + 3\ln x} \right)}^2}}}{x}dx} \hfill \\
Let\,\,u = 1 + 3\ln x\,\,,\,\,So\,\,that \hfill \\
du = \frac{3}{x}dx \hfill \\
Then \hfill \\
\frac{1}{3}\int_{}^{} {\frac{{3\,{{\left( {1 + 3\ln x} \right)}^2}}}{x}dx = \frac{1}{3}\int_{}^{} {{u^2}du} } \hfill \\
Integrating \hfill \\
\frac{1}{3}\,\left( {\frac{{{u^3}}}{3}} \right) + C \hfill \\
\frac{{{u^3}}}{9} + C \hfill \\
Substituting\,\,u = 1 + 3\ln x\,\,,\,\,gives \hfill \\
\frac{{\,{{\left( {1 + 3\ln x} \right)}^3}}}{9} + C \hfill \\
\end{gathered} \]
