Answer
$$ - \frac{2}{3}\left( {x + 4} \right)\sqrt {2 - x} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {2 - x} }}} dx \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{Rewrite the integrand}} \cr
& = \int {\frac{x}{{\sqrt {2 + \left( { - 1} \right)x} }}} dx \cr
& {\text{The integrand has a expression in the form }}\sqrt {a + bu} {} \cr
& {\text{Use formula 105}} \cr
& \left( {105} \right):\,\,\,\,\int {\frac{{udu}}{{\sqrt {a + bu} }}} = \frac{2}{{3{b^2}}}\left( {bu - 2a} \right)\sqrt {a + bu} + C \cr
& {\text{let }}u = x,\,\,\,a = 2{\text{ and }}b = - 1 \cr
& \int {\frac{x}{{\sqrt {2 + \left( { - 1} \right)x} }}} dx = \frac{2}{{3{{\left( { - 1} \right)}^2}}}\left( {\left( { - 1} \right)x - 2\left( 2 \right)} \right)\sqrt {2 + \left( { - 1} \right)x} + C \cr
& {\text{simplifying}} \cr
& \int {\frac{x}{{\sqrt {2 + \left( { - 1} \right)x} }}} dx = \frac{2}{3}\left( { - x - 4} \right)\sqrt {2 - x} + C \cr
& \int {\frac{x}{{\sqrt {2 + \left( { - 1} \right)x} }}} dx = - \frac{2}{3}\left( {x + 4} \right)\sqrt {2 - x} + C \cr} $$
