`int_(3)^(6) 2[x] dx` is equal to

Options

A. 12
B. 30
C. 18
D. 24

Answer

To solve the integral \( \int_{3}^{6} 2[x] \, dx \), where \([x]\) is the greatest integer function (also known as the floor function), we will break the integral into segments based on the integer values of \(x\).

### Step 1: Identify the intervals
The greatest integer function \([x]\) takes constant integer values over intervals. For \(x\) in the range from 3 to 6, we can identify the following intervals:
- From \(3\) to \(4\), \([x] = 3\)
- From \(4\) to \(5\), \([x] = 4\)
- From \(5\) to \(6\), \([x] = 5\)

### Step 2: Break the integral
We can break the integral into three parts based on the intervals identified:
\[ \int_{3}^{6} 2[x] \, dx = \int_{3}^{4} 2 \cdot 3 \, dx + \int_{4}^{5} 2 \cdot 4 \, dx + \int_{5}^{6} 2 \cdot 5 \, dx \]

### Step 3: Calculate each integral
Now we will calculate each integral separately.

1. **First integral**:
\[ \int_{3}^{4} 2 \cdot 3 \, dx = 6 \int_{3}^{4} 1 \, dx = 6 [x]_{3}^{4} = 6(4 - 3) = 6 \cdot 1 = 6 \]

2. **Second integral**:
\[ \int_{4}^{5} 2 \cdot 4 \, dx = 8 \int_{4}^{5} 1 \, dx = 8 [x]_{4}^{5} = 8(5 - 4) = 8 \cdot 1 = 8 \]

3. **Third integral**:
\[ \int_{5}^{6} 2 \cdot 5 \, dx = 10 \int_{5}^{6} 1 \, dx = 10 [x]_{5}^{6} = 10(6 - 5) = 10 \cdot 1 = 10 \]

### Step 4: Sum the results
Now we sum the results of the three integrals:
\[ \int_{3}^{6} 2[x] \, dx = 6 + 8 + 10 = 24 \]

### Final Answer
Thus, the value of the integral \( \int_{3}^{6} 2[x] \, dx \) is \( \boxed{24} \).