Answer
\[ = - \frac{1}{9}\ln \left| {\frac{{9 + \sqrt {81 - {x^2}} }}{x}} \right| + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{dx}}{{x\sqrt {81 - {x^2}} }}} \hfill \\
\hfill \\
use\,\,\int_{}^{} {\frac{{du}}{{u\sqrt {{a^2} - {u^2}} }}} = \frac{1}{a}\ln \left| {\frac{{a + \sqrt {{a^2} - {u^2}} }}{u}} \right| + C \hfill \\
\hfill \\
with\,\,a = 9 \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= - \frac{1}{9}\ln \left| {\frac{{9 + \sqrt {81 - {x^2}} }}{x}} \right| + C \hfill \\
\end{gathered} \]
