Question

Given that S^3 on the bottom 1 (e^x)dx=(e^3)-e use the properties of integrals and this result to evaluate S^3 on the bottom 1 (5e^x − 1) dx. I got 5e^3x-5e-1 is this correct? Please help!

Answer 1

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

$\int\limits^3_1 {5e^x - 1} \, dx$
$= \int\limits^3_1 {5e^x} \, dx - \int\limits^3_1 1} \, dx$
$= 5 \int\limits^3_1 {e^x} \, dx - \int\limits^3_1 1} \, dx$
$= 5(e^3 - e) - \left.x\right|_1^3$
$= 5e^3 - 5e - (3 - 1)$
$= \bf 5e^3 - 5e - 2$