`int_(0)^(3) [x]dx` is equal to

Options

A. 2
B. 4
C. 3
D. 1

Explanation

To solve the integral \( I = \int_{0}^{3} [x] \, dx \), 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. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, we can analyze the function in intervals:
- For \(0 \leq x < 1\), \([x] = 0\)
- For \(1 \leq x < 2\), \([x] = 1\)
- For \(2 \leq x < 3\), \([x] = 2\)

2. **Breaking the Integral**:
We can break the integral from \(0\) to \(3\) into three parts:
\[ I = \int_{0}^{1} [x] \, dx + \int_{1}^{2} [x] \, dx + \int_{2}^{3} [x] \, dx \]

3. **Evaluating Each Integral**:
- For the first part:
\[ \int_{0}^{1} [x] \, dx = \int_{0}^{1} 0 \, dx = 0 \]
- For the second part:
\[ \int_{1}^{2} [x] \, dx = \int_{1}^{2} 1 \, dx = 1 \cdot (2 - 1) = 1 \]
- For the third part:
\[ \int_{2}^{3} [x] \, dx = \int_{2}^{3} 2 \, dx = 2 \cdot (3 - 2) = 2 \]

4. **Combining the Results**:
Now, we can combine the results from all three parts:
\[ I = 0 + 1 + 2 = 3 \]

### Final Answer:
Thus, the value of the integral \( \int_{0}^{3} [x] \, dx \) is equal to \( 3 \).