Answer
$$\frac{x}{2}\sqrt {9 - {x^2}} + \frac{9}{2}{\sin ^{ - 1}}\frac{x}{3} + C$$
Work Step by Step
$$\eqalign{
& \int {\sqrt {9 - {x^2}} } dx \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{Rewrite the integrand}} \cr
& \int {\sqrt {9 - {x^2}} } dx = \int {\sqrt {{3^2} - {x^2}} } dx \cr
& {\text{The integrand has a expression in the form }}\sqrt {{a^2} - {u^2}} {} \cr
& {\text{Use formula 74}} \cr
& \left( {90} \right):\,\,\,\,\int {\sqrt {{a^2} - {u^2}} du} = \frac{u}{2}\sqrt {{a^2} - {u^2}} + \frac{{{a^2}}}{2}{\sin ^{ - 1}}\frac{u}{a} + C\,\,\, \cr
& {\text{let }}u = x,\,\,\,a = 3 \cr
& \int {\sqrt {{3^2} - {x^2}} } dx = \frac{x}{2}\sqrt {{3^2} - {x^2}} + \frac{{{{\left( 3 \right)}^2}}}{2}{\sin ^{ - 1}}\frac{x}{3} + C\,\,\, \cr
& {\text{simplifying}} \cr
& \int {\sqrt {9 - {x^2}} } dx = \frac{x}{2}\sqrt {9 - {x^2}} + \frac{9}{2}{\sin ^{ - 1}}\frac{x}{3} + C \cr} $$
