`int_0^1.5 [x] dx=___` where [x] is greatest integer function.

Answer

To solve the integral \( \int_0^{1.5} [x] \, dx \), where \([x]\) is the greatest integer function (also known as the floor function), we can break the integral into segments based on the behavior of the greatest integer function.

### Step-by-Step Solution:

1. **Identify the intervals**: The greatest integer function \([x]\) takes constant integer values over specific intervals. For \(x\) in the interval \([0, 1)\), \([x] = 0\). For \(x\) in the interval \([1, 1.5)\), \([x] = 1\).

2. **Break the integral**: We can split the integral into two parts:
\[ \int_0^{1.5} [x] \, dx = \int_0^1 [x] \, dx + \int_1^{1.5} [x] \, dx \]

3. **Evaluate the first integral**:
- For \(x\) in \([0, 1)\), \([x] = 0\).
- Therefore,
\[ \int_0^1 [x] \, dx = \int_0^1 0 \, dx = 0 \]

4. **Evaluate the second integral**:
- For \(x\) in \([1, 1.5)\), \([x] = 1\).
- Therefore,
\[ \int_1^{1.5} [x] \, dx = \int_1^{1.5} 1 \, dx \]
- This integral evaluates to:
\[ \int_1^{1.5} 1 \, dx = [x]_{1}^{1.5} = 1.5 - 1 = 0.5 \]

5. **Combine the results**:
\[ \int_0^{1.5} [x] \, dx = 0 + 0.5 = 0.5 \]

### Final Answer:
\[ \int_0^{1.5} [x] \, dx = 0.5 \]