`int _(0) ^(3) e ^(x) dx = `

Options

A. `e ^(3) -1 `
B. `1- e ^(3)`
C. `e -1`
D. `1-e`

Answer

To solve the integral \( \int_{0}^{3} e^{x} \, dx \), we will follow these steps:

### Step 1: Set up the integral
We start with the integral:
\[ I = \int_{0}^{3} e^{x} \, dx \]

### Step 2: Find the antiderivative of \( e^{x} \)
The antiderivative of \( e^{x} \) is:
\[ \int e^{x} \, dx = e^{x} + C \]
where \( C \) is the constant of integration.

### Step 3: Evaluate the definite integral
Now we will evaluate the definite integral from 0 to 3:
\[ I = \left[ e^{x} \right]_{0}^{3} = e^{3} - e^{0} \]

### Step 4: Simplify the expression
We know that \( e^{0} = 1 \), so:
\[ I = e^{3} - 1 \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{3} e^{x} \, dx = e^{3} - 1 \]

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