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`