Remember that
1) For functions [tex]f(x)[/tex] and [tex]g(x)[/tex], [tex]\int {f(x) + g(x)} \, dx = \int {f(x)} \, dx + \int {g(x)} \, dx [/tex]
2) For a function [tex]f(x)[/tex] and a constant [tex]c[/tex], [tex] \int {cf(x)} \, dx = c \int {f(x)} \, dx [/tex]
Using these two properties of integrals, and the fact that [tex] \int\limits^3_1 {e^x} \, dx = e^3 - e[/tex], we can see that
[tex]\int\limits^3_1 {5e^x - 1} \, dx[/tex]
[tex]= \int\limits^3_1 {5e^x} \, dx - \int\limits^3_1 1} \, dx[/tex]
[tex]= 5 \int\limits^3_1 {e^x} \, dx - \int\limits^3_1 1} \, dx[/tex]
[tex]= 5(e^3 - e) - \left.x\right|_1^3[/tex]
[tex]= 5e^3 - 5e - (3 - 1)[/tex]
[tex]= \bf 5e^3 - 5e - 2[/tex]
