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

Explanation

To solve the integral \( \int_{0}^{2} e^{x} \, dx \), we will follow the steps of integration directly without using the limit of sums approach mentioned in the video transcript.

### Step-by-Step Solution:

1. **Identify the Integral**: We need to evaluate the integral:
\[ \int_{0}^{2} e^{x} \, dx \]

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

3. **Evaluate the Definite Integral**: We will now evaluate the definite integral from 0 to 2:
\[ \int_{0}^{2} e^{x} \, dx = \left[ e^{x} \right]_{0}^{2} \]

4. **Substitute the Limits**: Now we substitute the upper limit and the lower limit:
\[ = e^{2} - e^{0} \]

5. **Simplify**: We know that \( e^{0} = 1 \), so:
\[ = e^{2} - 1 \]

6. **Final Result**: Thus, the value of the integral is:
\[ \int_{0}^{2} e^{x} \, dx = e^{2} - 1 \]