Answer
$\frac{1}{{23}}{\left( {x + 1} \right)^{23}} + \frac{1}{{11}}{\left( {x + 1} \right)^{22}} + \frac{1}{{21}}{\left( {x + 1} \right)^{21}} + C$
Work Step by Step
$$\eqalign{
& \int {{{\left( {x + 2} \right)}^2}{{\left( {x + 1} \right)}^{20}}} dx \cr
& {\text{Let }}u = x + 1,{\text{ }}x = u - 1,{\text{ }}dx = du \cr
& {\text{Applying the substitution}}{\text{, we obtain}} \cr
& = \int {{{\left( {u - 1 + 2} \right)}^2}{u^{20}}} du \cr
& = \int {{{\left( {u + 1} \right)}^2}{u^{20}}} du \cr
& {\text{Expanding the binomial}} \cr
& = \int {\left( {{u^2} + 2u + 1} \right){u^{20}}} du \cr
& {\text{Multiply}} \cr
& = \int {\left( {{u^{22}} + 2{u^{21}} + {u^{20}}} \right)} du \cr
& {\text{Integrate by using the power rule}} \cr
& = \frac{1}{{23}}{u^{23}} + \frac{1}{{11}}{u^{22}} + \frac{1}{{21}}{u^{21}} + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}x + 1{\text{ for }}u \cr
& = \frac{1}{{23}}{\left( {x + 1} \right)^{23}} + \frac{1}{{11}}{\left( {x + 1} \right)^{22}} + \frac{1}{{21}}{\left( {x + 1} \right)^{21}} + C \cr} $$
