Answer
$$630y$$
Work Step by Step
$$\eqalign{
& \int_0^5 {\left( {{x^4}y + y} \right)dx} \cr
& {\text{the notation }}dx{\text{ indicates integration with respect to }}x,{\text{ so we treat }}x{\text{ as variable and}} \cr
& y{\text{ as a constant}}{\text{. Using the power rule for antiderivatives we obtain}} \cr
& \int_0^5 {\left( {{x^4}y + y} \right)dx} = \left[ {y\left( {\frac{{{x^5}}}{5}} \right) + y\left( x \right)} \right]_0^5 \cr
& {\text{then}} \cr
& = \left[ {\frac{{{x^5}y}}{5} + xy} \right]_0^5 \cr
& {\text{evaluating the limits in the variable }}x \cr
& = \left[ {\frac{{{{\left( 5 \right)}^5}y}}{5} + \left( 5 \right)y} \right] - \left[ {\frac{{{{\left( 0 \right)}^5}y}}{5} + \left( 0 \right)y} \right] \cr
& {\text{simplifying}} \cr
& = {5^4}y + 5y \cr
& = 625y + 5y \cr
& = 630y \cr} $$