Answer
$$4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C$$
Work Step by Step
$$\eqalign{
& \int {{{\left( {2 + {y^2}} \right)}^2}dy} \cr
& {\text{expand integrand}} \cr
& = \int {\left( {{2^2} + 2\left( 2 \right)\left( {{y^2}} \right) + {{\left( {{y^2}} \right)}^2}} \right)dy} \cr
& = \int {\left( {4 + 4{y^2} + {y^4}} \right)dy} \cr
& = \int {4dy} + \int {4{y^2}} dy + \int {{y^4}} dy \cr
& {\text{power rule}} \cr
& = 4y + \frac{{4{y^{2 + 1}}}}{{2 + 1}} + \frac{{{y^{4 + 1}}}}{{4 + 1}} + C \cr
& = 4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C \cr
& \cr
& {\text{check by differentiation}} \cr
& = \frac{d}{{dx}}\left[ {4y + \frac{{4{y^3}}}{3} + \frac{{{y^5}}}{5} + C} \right] \cr
& = 4 + \frac{{4\left( {3{y^2}} \right)}}{3} + \frac{{5{y^4}}}{5} + C \cr
& = 4 + 4{y^2} + {y^4} \cr
& = {\left( {2 + y} \right)^2} \cr} $$
