Evaluate the following integrals:
`int_(-1)^(1)e^(|x|)dx`

Answer

To evaluate the integral \( \int_{-1}^{1} e^{|x|} \, dx \), we will break it down into two parts based on the definition of the absolute value function.

### Step 1: Understand the Absolute Value Function
The absolute value function \( |x| \) is defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Split the Integral
Given the limits of integration from \(-1\) to \(1\), we can split the integral at \(0\):
\[ \int_{-1}^{1} e^{|x|} \, dx = \int_{-1}^{0} e^{|x|} \, dx + \int_{0}^{1} e^{|x|} \, dx \]

### Step 3: Evaluate Each Integral
1. For the first integral, where \( x \) is in the interval \([-1, 0]\):
\[ |x| = -x \quad \text{(since \( x < 0 \))} \]
Thus,
\[ \int_{-1}^{0} e^{|x|} \, dx = \int_{-1}^{0} e^{-x} \, dx \]

2. For the second integral, where \( x \) is in the interval \([0, 1]\):
\[ |x| = x \quad \text{(since \( x \geq 0 \))} \]
Thus,
\[ \int_{0}^{1} e^{|x|} \, dx = \int_{0}^{1} e^{x} \, dx \]

### Step 4: Compute the Integrals
1. Compute \( \int_{-1}^{0} e^{-x} \, dx \):
\[ \int e^{-x} \, dx = -e^{-x} + C \]
Evaluating from \(-1\) to \(0\):
\[ \left[-e^{-x}\right]_{-1}^{0} = \left[-e^{0}\right] - \left[-e^{1}\right] = -1 + e = e - 1 \]

2. Compute \( \int_{0}^{1} e^{x} \, dx \):
\[ \int e^{x} \, dx = e^{x} + C \]
Evaluating from \(0\) to \(1\):
\[ \left[e^{x}\right]_{0}^{1} = e^{1} - e^{0} = e - 1 \]

### Step 5: Combine the Results
Now, we combine the results of both integrals:
\[ \int_{-1}^{1} e^{|x|} \, dx = (e - 1) + (e - 1) = 2(e - 1) \]

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