Evaluate the following
`int_(-1)^(1)|x|dx`.

Answer

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

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

### Step 2: Split the Integral
Since the absolute value function changes at \( x = 0 \), we will split the integral into two parts:
\[ \int_{-1}^{1} |x| \, dx = \int_{-1}^{0} |x| \, dx + \int_{0}^{1} |x| \, dx \]

### Step 3: Evaluate Each Integral
1. For the first integral, where \( x \) is from \(-1\) to \(0\):
\[ \int_{-1}^{0} |x| \, dx = \int_{-1}^{0} -x \, dx \]
Now, we can compute this integral:
\[ = -\int_{-1}^{0} x \, dx \]
The integral of \( x \) is:
\[ = -\left[\frac{x^2}{2}\right]_{-1}^{0} = -\left(0 - \frac{(-1)^2}{2}\right) = -\left(0 - \frac{1}{2}\right) = \frac{1}{2} \]

2. For the second integral, where \( x \) is from \(0\) to \(1\):
\[ \int_{0}^{1} |x| \, dx = \int_{0}^{1} x \, dx \]
Now, we can compute this integral:
\[ = \left[\frac{x^2}{2}\right]_{0}^{1} = \left(\frac{1^2}{2} - 0\right) = \frac{1}{2} \]

### Step 4: Combine the Results
Now, we can combine the results of the two integrals:
\[ \int_{-1}^{1} |x| \, dx = \int_{-1}^{0} |x| \, dx + \int_{0}^{1} |x| \, dx = \frac{1}{2} + \frac{1}{2} = 1 \]

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