Definite Integrals of Exponential Functions

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Question 1 of 2

1. Question

Find the integral

$$\int_{-1}^{3} e^{-\frac{x}{2}} dx$$

`-2/(sqrt(e^3)) +2sqrte`
`-2/(sqrt(e^3)) +2e`
`2/(sqrt(e^3)) +2sqrte`
`e^2-1`

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Integrating Exponential Functions with Base “e”

$$\int e^{\color{#004ec4}{a}x+b} dx=\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$

Substitute the components into the formula

$$\int e^{\color{#004ec4}{a}x+b} dx$$	`=`	$$\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$

$$\int_{-1}^{3} e^{\color{#004ec4}{-\frac{x}{2}}} dx$$	`=`	$$\int_{-1}^{3} \frac{e^{\color{#004ec4}{-\frac{x}{2}}}}{-\frac{1}{2}}$$	Substitute known values

	`=`	`[-2e^(-x/2)]_(-1)^3`	Simplify

Finally, get the difference of the upper and lower limits substituted to the integral as `x`.

	`[-2e^(-x/2)]_(-1)^3`

`=`	`[-2e^(-3/2)]-[-2e^(- (-1)/2)]`	Substitute the limits

`=`	`-2e^(-3/2)+2e^(1/2)`	Evaluate

`=`	`-2/e^(3/2) +2e^(1/2)`	Reciprocate `e^(-3/2)`

`=`	`-2/(sqrt(e^3)) +2sqrte`	Change the exponents into surds

`-2/(sqrt(e^3)) +2sqrte`

Question 2 of 2

2. Question