The value of `int_(0)^(2)e^(x)` is:

Options

A. e
B. 1
C. `1/(e)`
D. `e^(2)-1`

Answer

To solve the integral \( \int_{0}^{2} e^{x} \, dx \), we will follow these steps:

### Step 1: Identify the integral
We need to evaluate the definite integral of \( e^{x} \) from \( 0 \) to \( 2 \).

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

### Step 3: Evaluate the definite integral
Using the Fundamental Theorem of Calculus, we can evaluate the definite integral:
\[ \int_{0}^{2} e^{x} \, dx = \left[ e^{x} \right]_{0}^{2} \]
This means we will calculate \( e^{2} \) and \( e^{0} \) and then find the difference.

### Step 4: Substitute the limits
Now we substitute the limits into the antiderivative:
\[ = e^{2} - e^{0} \]

### Step 5: Simplify the expression
Since \( e^{0} = 1 \), we have:
\[ = e^{2} - 1 \]

### Final Answer
Thus, the value of the integral \( \int_{0}^{2} e^{x} \, dx \) is:
\[ e^{2} - 1 \]