The value of the integral `int_(-1)^(1) (x-[2x])`dx,is

Options

A. 1
B. 0
C. 2
D. 4

Answer

To solve the integral \( I = \int_{-1}^{1} (x - [2x]) \, dx \), where \([2x]\) denotes the greatest integer function, we will break down the integral into manageable parts based on the behavior of the greatest integer function.

### Step 1: Split the Integral
We can separate the integral into two parts:
\[ I = \int_{-1}^{1} x \, dx - \int_{-1}^{1} [2x] \, dx \]

### Step 2: Calculate the First Integral
The first integral is straightforward:
\[ \int_{-1}^{1} x \, dx = \left[ \frac{x^2}{2} \right]_{-1}^{1} = \frac{1^2}{2} - \frac{(-1)^2}{2} = \frac{1}{2} - \frac{1}{2} = 0 \]

### Step 3: Analyze the Second Integral
Next, we need to evaluate \( \int_{-1}^{1} [2x] \, dx \). The function \([2x]\) changes its value at certain points in the interval \([-1, 1]\).

1. For \( x \in [-1, -0.5) \):
- \( 2x \in [-2, -1) \)
- \([2x] = -2\)

2. For \( x \in [-0.5, 0) \):
- \( 2x \in [-1, 0) \)
- \([2x] = -1\)

3. For \( x \in [0, 0.5) \):
- \( 2x \in [0, 1) \)
- \([2x] = 0\)

4. For \( x \in [0.5, 1] \):
- \( 2x \in [1, 2] \)
- \([2x] = 1\)

### Step 4: Break Down the Integral
Now we can break down the integral \( \int_{-1}^{1} [2x] \, dx \) into four parts:
\[ \int_{-1}^{-0.5} -2 \, dx + \int_{-0.5}^{0} -1 \, dx + \int_{0}^{0.5} 0 \, dx + \int_{0.5}^{1} 1 \, dx \]

### Step 5: Calculate Each Part
1. \( \int_{-1}^{-0.5} -2 \, dx = -2 \left[ x \right]_{-1}^{-0.5} = -2 \left( -0.5 + 1 \right) = -2 \cdot 0.5 = -1 \)

2. \( \int_{-0.5}^{0} -1 \, dx = -1 \left[ x \right]_{-0.5}^{0} = -1 \left( 0 + 0.5 \right) = -0.5 \)

3. \( \int_{0}^{0.5} 0 \, dx = 0 \)

4. \( \int_{0.5}^{1} 1 \, dx = \left[ x \right]_{0.5}^{1} = 1 - 0.5 = 0.5 \)

### Step 6: Combine the Results
Now, we can combine the results of the second integral:
\[ \int_{-1}^{1} [2x] \, dx = -1 - 0.5 + 0 + 0.5 = -1 \]

### Step 7: Final Calculation
Now we can substitute this back into our expression for \( I \):
\[ I = 0 - (-1) = 1 \]

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