Answer
$- \frac{{{{\left( {5 - 3x} \right)}^{11}}}}{{33}} + C$
Work Step by Step
$$\eqalign{
& \int {{{\left( {5 - 3x} \right)}^{10}}} dx \cr
& {\text{Let }}u = 5 - 3x,{\text{ then }}du = - 3dx,{\text{ }}dx = - \frac{1}{3}du \cr
& {\text{Applying the substitution}}{\text{, we obtain}} \cr
& \int {{{\left( {5 - 3x} \right)}^{10}}} dx = \int {{u^{10}}} \left( { - \frac{1}{3}} \right)du \cr
& = - \frac{1}{3}\int {{u^{10}}} du \cr
& {\text{Integrate apply the power rule }}\int {{u^n}du} = \frac{{{u^{n + 1}}}}{{n + 1}}{\text{ + C }} \cr
& = - \frac{1}{3}\left( {\frac{{{u^{11}}}}{{11}}} \right) + C \cr
& = - \frac{{{u^{11}}}}{{33}} + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}u = 5 - 3x \cr
& = - \frac{{{{\left( {5 - 3x} \right)}^{11}}}}{{33}} + C \cr} $$