Let `[x]` denote the greatest integer less than or equal to x, then the value of the integral `int_(-1)^(1)(|x|-2[x])dx` is equal to

Options

A. 3
B. 2
C. `-2`
D. `-3`

Answer

To solve the integral \( I = \int_{-1}^{1} (|x| - 2[x]) \, dx \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we can break the integral into two parts based on the behavior of the functions involved.

### Step 1: Analyze the function
The absolute value function \(|x|\) behaves differently for negative and positive values of \(x\):
- For \(x < 0\), \(|x| = -x\)
- For \(x \geq 0\), \(|x| = x\)

The greatest integer function \([x]\) also behaves differently:
- For \(-1 \leq x < 0\), \([x] = -1\)
- For \(0 \leq x < 1\), \([x] = 0\)

### Step 2: Split the integral
We can split the integral into two parts:
\[ I = \int_{-1}^{0} (|x| - 2[x]) \, dx + \int_{0}^{1} (|x| - 2[x]) \, dx \]

### Step 3: Evaluate the first integral from \(-1\) to \(0\)
For \(x\) in the interval \([-1, 0)\):
- \(|x| = -x\)
- \([x] = -1\)

Thus, the integral becomes:
\[ \int_{-1}^{0} (-x - 2(-1)) \, dx = \int_{-1}^{0} (-x + 2) \, dx \]

### Step 4: Calculate the first integral
Now, we compute:
\[ \int_{-1}^{0} (-x + 2) \, dx = \int_{-1}^{0} (-x) \, dx + \int_{-1}^{0} 2 \, dx \]
Calculating each part:
1. \(\int_{-1}^{0} -x \, dx = \left[-\frac{x^2}{2}\right]_{-1}^{0} = 0 - \frac{(-1)^2}{2} = -\frac{1}{2}\)
2. \(\int_{-1}^{0} 2 \, dx = 2[x]_{-1}^{0} = 2(0 - (-1)) = 2\)

Combining these results:
\[ \int_{-1}^{0} (-x + 2) \, dx = -\frac{1}{2} + 2 = \frac{3}{2} \]

### Step 5: Evaluate the second integral from \(0\) to \(1\)
For \(x\) in the interval \([0, 1)\):
- \(|x| = x\)
- \([x] = 0\)

Thus, the integral becomes:
\[ \int_{0}^{1} (x - 2(0)) \, dx = \int_{0}^{1} x \, dx \]

### Step 6: Calculate the second integral
Calculating this integral:
\[ \int_{0}^{1} x \, dx = \left[\frac{x^2}{2}\right]_{0}^{1} = \frac{1^2}{2} - 0 = \frac{1}{2} \]

### Step 7: Combine both integrals
Now, we combine the results of both integrals:
\[ I = \frac{3}{2} + \frac{1}{2} = 2 \]

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