How To Integrate sqrt[9-x^2]/x^2 by Trigonometric Substitution

 

\begin{array}{l} \smallint \frac{{\sqrt {{\bf{9}} - {x^{\bf{2}}}} }}{{{x^{\bf{2}}}}}{\rm{d}}x\\\\ Let \Rightarrow 1 \cdot x = 3\sin \theta \;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{d}}x = 3cos\theta {\rm{d}}\theta \\ Change{\rm{ }}the\;expression\;\;\;\;\;\;\\ \;\;\;\;\;{x^2} = 9{\sin ^2}\theta \;\;\;\;\;\;\;\\ \;\;\;\;9 - {x^2} = 9 - 9{\sin ^2}\theta \;\;\;\;\;\;\;\\ \;\;\;\;9 - {x^2} = 9\left( {1 - {{\sin }^2}\theta } \right)\;\;\;\;\;\;\\ \;\;\;\;\;9 - {x^2} = 9{\cos ^2}\theta \;\;\;\;\;\;\\ \;\;\;\;\;\sqrt {9 - {x^2}} = 3\cos \theta \\ \smallint \frac{{3\cos \theta }}{{9{{\sin }^2}\theta }} \cdot 3\cos \theta {\rm{d}}\theta = \smallint \frac{{co{s^2}\theta }}{{si{n^2}\theta }}{\rm{d}}\theta \\ \smallint {\cot ^2}\theta {\rm{d}}\theta = \smallint cs{c^2}\theta - 1{\rm{d}}\theta \\ \Rightarrow - \cot \theta - \theta + c \end{array}

\begin{array}{l} \Rightarrow - \frac{{\sqrt {9 - {x^2}} }}{x} - {\sin ^{ - 1}}\frac{x}{2} + c \end{array}