`int_(-1)^(1)e^(|x|)dx=2(e-1)`

Explanation

To solve 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: Break the integral at the point where the absolute value changes
The absolute value function \( |x| \) can be expressed as:
- \( |x| = -x \) when \( x < 0 \)
- \( |x| = x \) when \( x \geq 0 \)

Thus, we can split the integral as follows:
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