The value of ` int_(0)^(2)[x+[x+[x]]] dx` (where, [.] denotes the greatest integer function )is equal to

Options

A. 2
B. 3
C. `-3`
D. None of these

Answer

To solve the integral \( \int_{0}^{2} [x + [x + [x]]] \, dx \), where \([.]\) denotes the greatest integer function, we can break down the problem step by step.

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). We will analyze the function \( [x + [x + [x]]] \) over the interval \([0, 2]\).

### Step 2: Determine the intervals
We will consider the intervals where the function \([x]\) changes:
- For \(0 \leq x < 1\), \([x] = 0\)
- For \(1 \leq x < 2\), \([x] = 1\)

### Step 3: Evaluate the function in the intervals
1. **For \(0 \leq x < 1\)**:
- \([x] = 0\)
- \([x + [x]] = [x + 0] = [x] = 0\)
- \([x + [x + [x]]] = [x + 0] = [x] = 0\)
- Therefore, \( [x + [x + [x]]] = 0 \).

2. **For \(1 \leq x < 2\)**:
- \([x] = 1\)
- \([x + [x]] = [x + 1] = [x] + 1 = 1 + 1 = 2\)
- \([x + [x + [x]]] = [x + 2] = [x] + 2 = 1 + 2 = 3\)
- Therefore, \( [x + [x + [x]]] = 3 \).

### Step 4: Set up the integral
Now we can set up the integral based on the intervals:
\[ \int_{0}^{2} [x + [x + [x]]] \, dx = \int_{0}^{1} 0 \, dx + \int_{1}^{2} 3 \, dx \]

### Step 5: Calculate the integrals
1. **First integral**:
\[ \int_{0}^{1} 0 \, dx = 0 \]

2. **Second integral**:
\[ \int_{1}^{2} 3 \, dx = 3 \cdot (2 - 1) = 3 \]

### Step 6: Combine the results
Adding the results from both integrals:
\[ \int_{0}^{2} [x + [x + [x]]] \, dx = 0 + 3 = 3 \]

### Final Answer
Thus, the value of the integral is \( \boxed{3} \).