Answer
\[ = - 3\ln \left| {\frac{{3 + \sqrt {9 - {x^2}} }}{x}} \right| + \sqrt {9 - {x^2}} + C\]
Work Step by Step
\[\begin{gathered}
\int_{}^{} {\frac{{\sqrt {9 - {x^2}} }}{x}} \,\,dx \hfill \\
\hfill \\
the\,\,integral\,\,has\,\,the\,\,form\,\,{a^2} - {x^2}\,\, \hfill \\
\hfill \\
x = 3\sin \theta \,\,\,\,\,\,then\,\,\,\,\,dx = 3\cos \theta d\theta \hfill \\
and\,\,\sqrt {9 - {x^2}} = 3\cos \theta \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= \int_{}^{} {\frac{{\sqrt {9 - {x^2}} }}{x}dx} \hfill \\
\hfill \\
= \int_{}^{} {\frac{{\,\left( {3\cos \theta } \right)}}{{\,\left( {3\sin \theta } \right)}}\,\left( {3\cos \theta } \right)d\theta } \hfill \\
\hfill \\
= 3\int_{}^{} {\frac{{{{\cos }^2}\theta }}{{\sin \theta }}d\theta \,} \hfill \\
\hfill \\
= 3\int_{}^{} {\frac{{\,\left( {1 - {{\sin }^2}\theta } \right)}}{{\sin \theta }}d\theta } \hfill \\
\hfill \\
distribute \hfill \\
\hfill \\
= 3\int_{}^{} {\csc d\theta } - 3\int_{}^{} {\sin \theta d\theta } \hfill \\
\hfill \\
integrate \hfill \\
\hfill \\
= - \ln \left| {\csc \,\,\,\theta + \cot \,\,\theta } \right| + 3\cos \,\,\theta + C \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= - 3\ln \left| {\frac{{3 + \sqrt {9 - {x^2}} }}{x}} \right| + \sqrt {9 - {x^2}} + C \hfill \\
\hfill \\
\end{gathered} \]
