Answer
$$ - \sqrt {9 - {x^2}} + C $$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {9 - {x^2}} }}} dx \cr
& {\text{Integrate by substitution method}} \cr
& {\text{Let }}u = 9 - {x^2},\,\,\,\,du = - 2xdx,\,\,\,\,dx = \frac{{du}}{{ - 2x}} \cr
& {\text{Then}}{\text{,}} \cr
& \int {\frac{x}{{\sqrt {9 - {x^2}} }}} dx = \int {\frac{x}{{\sqrt u }}} \left( {\frac{{du}}{{ - 2x}}} \right) \cr
& {\text{Cancel common factor }}x \cr
& = \int {\frac{1}{{\sqrt u }}} \left( {\frac{{du}}{{ - 2}}} \right) \cr
& = - \frac{1}{2}\int {\frac{1}{{{u^{1/2}}}}du} \cr
& = - \frac{1}{2}\int {{u^{ - 1/2}}du} \cr
& {\text{integrating}} \cr
& = - \frac{1}{2}\left( {\frac{{{u^{1/2}}}}{{1/2}}} \right) + C \cr
& = - \sqrt u + C \cr
& {\text{write in terms of }}x.{\text{ }}u = 9 - {x^2} \cr
& = - \sqrt {9 - {x^2}} + C \cr} $$
