Answer
$$\frac{{{x^3}}}{3} + {x^2} + x + C$$
Work Step by Step
$$\eqalign{
& \int {{{\left( {x + 1} \right)}^2}} dx \cr
& {\text{expand the integrand using }}{\left( {A + B} \right)^2} = {A^2} + 2AB + {B^2} \cr
& = \int {\left( {{x^2} + 2x + 1} \right)} dx \cr
& {\text{use sum rule for integrals}} \cr
& = \int {{x^2}} dx + \int {2x} dx + \int {dx} \cr
& {\text{use }}\int {{x^n}dx} = \frac{{{x^{n + 1}}}}{{n + 1}} + C{\text{ and }}\int {dx} = x + C{\text{ then}} \cr
& = \frac{{{x^{2 + 1}}}}{{2 + 1}} + 2\left( {\frac{{{x^{1 + 1}}}}{{1 + 1}}} \right) + x + C \cr
& {\text{simplifying}} \cr
& = \frac{{{x^3}}}{3} + 2\left( {\frac{{{x^2}}}{2}} \right) + x + C \cr
& = \frac{{{x^3}}}{3} + {x^2} + x + C \cr} $$
