Let `[x]` denote the greatest integer less than or equal to `x`. Then, `int_(0)^(1.5)[x]dx=?`

Options

A. 6
B. 4
C. 1/2
D. 0

Answer

To solve the integral \( \int_{0}^{1.5} [x] \, dx \), 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 greatest integer function.

### Step 1: Identify the intervals
The function \([x]\) changes at integer values. Therefore, we can split the integral at the integer points:
- From \(0\) to \(1\)
- From \(1\) to \(1.5\)

### Step 2: Write the integral as a sum of two integrals
We can express the integral as:
\[ \int_{0}^{1.5} [x] \, dx = \int_{0}^{1} [x] \, dx + \int_{1}^{1.5} [x] \, dx \]

### Step 3: Evaluate the first integral \( \int_{0}^{1} [x] \, dx \)
In the interval \([0, 1)\), the greatest integer function \([x]\) is \(0\):
\[ \int_{0}^{1} [x] \, dx = \int_{0}^{1} 0 \, dx = 0 \]

### Step 4: Evaluate the second integral \( \int_{1}^{1.5} [x] \, dx \)
In the interval \([1, 1.5)\), the greatest integer function \([x]\) is \(1\):
\[ \int_{1}^{1.5} [x] \, dx = \int_{1}^{1.5} 1 \, dx \]
Calculating this integral:
\[ \int_{1}^{1.5} 1 \, dx = [x]_{1}^{1.5} = 1.5 - 1 = 0.5 \]

### Step 5: Combine the results
Now, we can combine the results of the two integrals:
\[ \int_{0}^{1.5} [x] \, dx = 0 + 0.5 = 0.5 \]

### Final Answer
Thus, the value of the integral \( \int_{0}^{1.5} [x] \, dx \) is:
\[ \boxed{0.5} \]