Answer
$$\frac{y}{4}\left( {{e^{20 + {y^2}}} - {e^{4 + {y^2}}}} \right)$$
Work Step by Step
$$\eqalign{
& \int_1^5 {y{e^{4x + {y^2}}}} dx \cr
& {\text{the notation }}dx{\text{ indicates integration with respect to }}x,{\text{ so we treat }}x{\text{ as a variable and}} \cr
& y{\text{ as a constant}}{\text{. then}} \cr
& \int_1^5 {y{e^{4x + {y^2}}}} dx = \frac{y}{4}\int_1^5 {{e^{4x + {y^2}}}\left( 4 \right)} dy \cr
& {\text{using }}\int {{e^u}} du = {e^u} + C \cr
& = \frac{y}{4}\left( {{e^{4x + {y^2}}}} \right)_1^5 \cr
& {\text{evaluating the limits in the variable }}x \cr
& = \frac{y}{4}\left( {{e^{4\left( 5 \right) + {y^2}}} - {e^{4\left( 1 \right) + {y^2}}}} \right) \cr
& {\text{simplifying}} \cr
& = \frac{y}{4}\left( {{e^{20 + {y^2}}} - {e^{4 + {y^2}}}} \right) \cr} $$
