Answer
$$6$$
Work Step by Step
$$\eqalign{
& \int_0^9 {\frac{{dx}}{{\sqrt {9 - x} }}} \cr
& {\text{The integrand is not defined for }}x = 9,{\text{ then}} \cr
& \int_0^9 {\frac{{dx}}{{\sqrt {9 - x} }}} = \mathop {\lim }\limits_{b \to {9^ - }} \int_0^b {\frac{{dx}}{{\sqrt {9 - x} }}} \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{b \to {9^ - }} \left[ { - 2\sqrt {9 - x} } \right]_0^b \cr
& = - 2\mathop {\lim }\limits_{b \to {9^ - }} \left[ {\sqrt {9 - b} - \sqrt {9 - 0} } \right] \cr
& = - 2\mathop {\lim }\limits_{b \to {9^ - }} \left[ {\sqrt {9 - b} - 3} \right] \cr
& {\text{Evaluating the limit when }}b \to {9^ - } \cr
& = - 2\left[ {\sqrt {9 - 9} - 3} \right] \cr
& = 6 \cr} $$
