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