` int _(-1)^(1)| 1 - x| dx ` is equal to

Options

A. `-2`
B. 0
C. 2
D. 4

Explanation

To solve the integral \( I = \int_{-1}^{1} |1 - x| \, dx \), we will break it down step by step.

### Step 1: Understand the Absolute Value Function
The expression \( |1 - x| \) changes based on the value of \( x \):
- When \( x < 1 \), \( 1 - x \) is positive, so \( |1 - x| = 1 - x \).
- When \( x = 1 \), \( |1 - x| = 0 \).
- When \( x > 1 \), \( |1 - x| = x - 1 \) (not applicable here since our limits are from -1 to 1).

Since we are integrating from -1 to 1, we only need to consider the case where \( x < 1 \).

### Step 2: Set Up the Integral
We can split the integral into two parts because the function \( |1 - x| \) is continuous and behaves differently at \( x = 1 \):
\[ I = \int_{-1}^{1} |1 - x| \, dx = \int_{-1}^{1} (1 - x) \, dx \]

### Step 3: Evaluate the Integral
Now we can evaluate the integral:
\[ I = \int_{-1}^{1} (1 - x) \, dx \]

This can be split into two separate integrals:
\[ I = \int_{-1}^{1} 1 \, dx - \int_{-1}^{1} x \, dx \]

### Step 4: Calculate Each Integral
1. Calculate \( \int_{-1}^{1} 1 \, dx \):
\[ \int_{-1}^{1} 1 \, dx = [x]_{-1}^{1} = 1 - (-1) = 2 \]

2. Calculate \( \int_{-1}^{1} x \, dx \):
\[ \int_{-1}^{1} x \, dx = \left[\frac{x^2}{2}\right]_{-1}^{1} = \frac{1^2}{2} - \frac{(-1)^2}{2} = \frac{1}{2} - \frac{1}{2} = 0 \]

### Step 5: Combine the Results
Now we can combine the results from the two integrals:
\[ I = 2 - 0 = 2 \]

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