Evaluate : `int _(-1) ^(1) e^(x) dx` .

Answer

To evaluate the integral \(\int_{-1}^{1} e^x \, dx\), we will follow these steps:

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

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

### Step 3: Apply the limits of integration
Now we will apply the limits from \(-1\) to \(1\):
\[ \int_{-1}^{1} e^x \, dx = \left[ e^x \right]_{-1}^{1} \]

### Step 4: Evaluate at the upper limit
First, we evaluate at the upper limit \(x = 1\):
\[ e^1 = e \]

### Step 5: Evaluate at the lower limit
Next, we evaluate at the lower limit \(x = -1\):
\[ e^{-1} = \frac{1}{e} \]

### Step 6: Subtract the lower limit from the upper limit
Now, we subtract the value at the lower limit from the value at the upper limit:
\[ \int_{-1}^{1} e^x \, dx = e - \frac{1}{e} \]

### Final Answer
Thus, the value of the integral is:
\[ e - \frac{1}{e} \]