If [.] stands for the greatest integer function, then `int_(1)^(2) [3x]dx` is equal to

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the integral \( \int_{1}^{2} [3x] \, dx \), where \([x]\) denotes the greatest integer function, we will break the integral into segments where the function \([3x]\) remains constant.

### Step 1: Determine the behavior of the function \([3x]\) over the interval \([1, 2]\)

1. Calculate \(3x\) at the endpoints of the interval:
- At \(x = 1\), \(3 \cdot 1 = 3\)
- At \(x = 2\), \(3 \cdot 2 = 6\)

2. Identify the points where \([3x]\) changes its value:
- The function \([3x]\) will change at \(x = \frac{4}{3}\) and \(x = \frac{5}{3}\) because:
- At \(x = \frac{4}{3}\), \(3 \cdot \frac{4}{3} = 4\)
- At \(x = \frac{5}{3}\), \(3 \cdot \frac{5}{3} = 5\)

### Step 2: Break the integral into segments

The integral can be split into three parts based on the intervals:
- From \(1\) to \(\frac{4}{3}\)
- From \(\frac{4}{3}\) to \(\frac{5}{3}\)
- From \(\frac{5}{3}\) to \(2\)

Thus, we can write:
\[ \int_{1}^{2} [3x] \, dx = \int_{1}^{\frac{4}{3}} [3x] \, dx + \int_{\frac{4}{3}}^{\frac{5}{3}} [3x] \, dx + \int_{\frac{5}{3}}^{2} [3x] \, dx \]

### Step 3: Evaluate each segment

1. **For \(x \in [1, \frac{4}{3})\)**:
- Here, \(3x\) ranges from \(3\) to just below \(4\), so \([3x] = 3\).
- Thus,
\[ \int_{1}^{\frac{4}{3}} [3x] \, dx = \int_{1}^{\frac{4}{3}} 3 \, dx = 3 \left( \frac{4}{3} - 1 \right) = 3 \cdot \frac{1}{3} = 1 \]

2. **For \(x \in [\frac{4}{3}, \frac{5}{3})\)**:
- Here, \(3x\) ranges from \(4\) to just below \(5\), so \([3x] = 4\).
- Thus,
\[ \int_{\frac{4}{3}}^{\frac{5}{3}} [3x] \, dx = \int_{\frac{4}{3}}^{\frac{5}{3}} 4 \, dx = 4 \left( \frac{5}{3} - \frac{4}{3} \right) = 4 \cdot \frac{1}{3} = \frac{4}{3} \]

3. **For \(x \in [\frac{5}{3}, 2]\)**:
- Here, \(3x\) ranges from \(5\) to \(6\), so \([3x] = 5\).
- Thus,
\[ \int_{\frac{5}{3}}^{2} [3x] \, dx = \int_{\frac{5}{3}}^{2} 5 \, dx = 5 \left( 2 - \frac{5}{3} \right) = 5 \cdot \frac{1}{3} = \frac{5}{3} \]

### Step 4: Combine the results

Now, we can add up the results from each segment:
\[ \int_{1}^{2} [3x] \, dx = 1 + \frac{4}{3} + \frac{5}{3} \]
Convert \(1\) to a fraction:
\[ 1 = \frac{3}{3} \]
Thus,
\[ \int_{1}^{2} [3x] \, dx = \frac{3}{3} + \frac{4}{3} + \frac{5}{3} = \frac{12}{3} = 4 \]

### Final Answer
\[ \int_{1}^{2} [3x] \, dx = 4 \]