Answer
$$\frac{{\sqrt 3 }}{{27}}$$
Work Step by Step
$$\eqalign{
& \int_0^{3/2} {\frac{{dx}}{{{{\left( {9 - {x^2}} \right)}^{3/2}}}}} \cr
& {\text{The integrand contains the form }}{a^2} - {x^2} \cr
& 9 - {x^2} \to a = 3 \cr
& {\text{Use the change of variable }}x = a\sin \theta \cr
& x = 3\sin \theta ,\,\,\,dx = 3\cos \theta d\theta \cr
& \cr
& {\text{Substituting}} \cr
& {\text{ = }}\int {\frac{{3\cos \theta d\theta }}{{{{\left( {9 - 9{{\sin }^2}\theta } \right)}^{3/2}}}}} \cr
& {\text{ = }}\int {\frac{{3\cos \theta }}{{{{\left( {9{{\cos }^2}\theta } \right)}^{3/2}}}}} d\theta \cr
& {\text{ = }}\int {\frac{{3\cos \theta }}{{27{{\cos }^3}\theta }}} d\theta \cr
& {\text{ = }}\frac{1}{9}\int {\frac{1}{{{{\cos }^2}\theta }}} d\theta \cr
& {\text{ = }}\frac{1}{9}\int {{{\sec }^2}\theta } d\theta \cr
& {\text{Integrating}} \cr
& {\text{ = }}\frac{1}{9}\tan \theta + C \cr
& {\text{Write in terms of }}x \cr
& {\text{ = }}\frac{1}{9}\left( {\frac{x}{{\sqrt {9 - {x^2}} }}} \right) + C \cr
& \cr
& ,{\text{then}} \cr
& \int_0^{3/2} {\frac{{dx}}{{{{\left( {9 - {x^2}} \right)}^{3/2}}}}} = \frac{1}{9}\left( {\frac{x}{{\sqrt {9 - {x^2}} }}} \right)_0^{3/2} \cr
& = \frac{1}{9}\left( {\frac{{3/2}}{{\sqrt {9 - {{\left( {3/2} \right)}^2}} }} - \frac{0}{{\sqrt {9 - {{\left( 0 \right)}^2}} }}} \right) \cr
& = \frac{1}{9}\left( {\frac{{\sqrt 3 }}{3} - 0} \right) \cr
& = \frac{{\sqrt 3 }}{{27}} \cr} $$
