Answer
$\frac{25}{4}\pi$
Work Step by Step
\[\begin{align}
& \int_{1}^{{{e}^{5}}}{\frac{\sqrt{25-{{\left( \ln x \right)}^{2}}}}{x}}dx \\
& \text{Let }u=\ln x,\text{ }du=\frac{1}{x}dx \\
& \text{The new limits of integration are:} \\
& x={{e}^{5}}\to u=5 \\
& x=1\to u=0 \\
& \text{Substituting} \\
& \int_{1}^{{{e}^{5}}}{\frac{\sqrt{25-{{\left( \ln x \right)}^{2}}}}{x}}dx=\int_{0}^{5}{\sqrt{25-{{u}^{2}}}}du \\
& \text{By the equation of a semicirle} \\
& \int_{0}^{5}{\sqrt{25-{{u}^{2}}}}du=\frac{1}{2}\left( \frac{1}{2}\pi {{\left( 5 \right)}^{2}} \right) \\
& =\frac{25}{4}\pi \\
\end{align}\]
