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