`int_(0)^(2) e^(x) dx`

Answer

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

### Step 1: Set up the integral
We start by defining the integral:
\[ I = \int_{0}^{2} e^{x} \, dx \]

### Step 2: Find the antiderivative
The antiderivative of \( e^{x} \) is \( e^{x} \). Therefore, we can express the integral as:
\[ I = \left[ e^{x} \right]_{0}^{2} \]

### Step 3: Evaluate the definite integral
Now we will evaluate the antiderivative at the upper and lower limits:
\[ I = e^{2} - e^{0} \]

### Step 4: Simplify the expression
Since \( e^{0} = 1 \), we can simplify the expression:
\[ I = e^{2} - 1 \]

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