`int_(0)^(5) [x] dx`= ….., where [x] denotes the greatest integer function

Options

A. 10
B. 8
C. 6
D. 4

Answer

To solve the integral \( \int_{0}^{5} [x] \, dx \), where \([x]\) denotes the greatest integer function (also known as the floor function), we can break the integral into segments based on the integer values of \(x\).

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example:
- For \(0 \leq x < 1\), \([x] = 0\)
- For \(1 \leq x < 2\), \([x] = 1\)
- For \(2 \leq x < 3\), \([x] = 2\)
- For \(3 \leq x < 4\), \([x] = 3\)
- For \(4 \leq x < 5\), \([x] = 4\)

2. **Breaking the Integral**:
We can break the integral from \(0\) to \(5\) into segments where \([x]\) is constant:
\[ \int_{0}^{5} [x] \, dx = \int_{0}^{1} [x] \, dx + \int_{1}^{2} [x] \, dx + \int_{2}^{3} [x] \, dx + \int_{3}^{4} [x] \, dx + \int_{4}^{5} [x] \, dx \]

3. **Evaluating Each Segment**:
- For \( \int_{0}^{1} [x] \, dx = \int_{0}^{1} 0 \, dx = 0\)
- For \( \int_{1}^{2} [x] \, dx = \int_{1}^{2} 1 \, dx = 1 \cdot (2 - 1) = 1\)
- For \( \int_{2}^{3} [x] \, dx = \int_{2}^{3} 2 \, dx = 2 \cdot (3 - 2) = 2\)
- For \( \int_{3}^{4} [x] \, dx = \int_{3}^{4} 3 \, dx = 3 \cdot (4 - 3) = 3\)
- For \( \int_{4}^{5} [x] \, dx = \int_{4}^{5} 4 \, dx = 4 \cdot (5 - 4) = 4\)

4. **Adding the Results**:
Now, we sum all these results:
\[ \int_{0}^{5} [x] \, dx = 0 + 1 + 2 + 3 + 4 = 10 \]

### Final Answer:
Thus, the value of the integral \( \int_{0}^{5} [x] \, dx \) is \( \boxed{10} \).