`int_(1)^(4) log_(e)[x]dx` equals

Options

A. `log_(e)6`
B. `log_(e)3`
C. `log_(e)2`
D. none of these

Explanation

To solve the definite integral \( \int_{1}^{4} \log_{e}[x] \, dx \), we first need to understand how the function behaves over the interval from 1 to 4, particularly because we are dealing with the integer part of \( x \).

### Step 1: Identify the intervals based on the integer part of \( x \)

The integer part function, denoted as \( \lfloor x \rfloor \), takes on constant values over intervals. For \( x \) in the range:
- From 1 to 2, \( \lfloor x \rfloor = 1 \)
- From 2 to 3, \( \lfloor x \rfloor = 2 \)
- From 3 to 4, \( \lfloor x \rfloor = 3 \)

Thus, we can break the integral into three parts:

\[ \int_{1}^{4} \log_{e}[\lfloor x \rfloor] \, dx = \int_{1}^{2} \log_{e}(1) \, dx + \int_{2}^{3} \log_{e}(2) \, dx + \int_{3}^{4} \log_{e}(3) \, dx \]

### Step 2: Evaluate each integral

1. **First Integral**:
\[ \int_{1}^{2} \log_{e}(1) \, dx = \int_{1}^{2} 0 \, dx = 0 \]

2. **Second Integral**:
\[ \int_{2}^{3} \log_{e}(2) \, dx = \log_{e}(2) \int_{2}^{3} 1 \, dx = \log_{e}(2) \cdot (3 - 2) = \log_{e}(2) \]

3. **Third Integral**:
\[ \int_{3}^{4} \log_{e}(3) \, dx = \log_{e}(3) \int_{3}^{4} 1 \, dx = \log_{e}(3) \cdot (4 - 3) = \log_{e}(3) \]

### Step 3: Combine the results

Now, we can combine the results of the three integrals:

\[ \int_{1}^{4} \log_{e}[\lfloor x \rfloor] \, dx = 0 + \log_{e}(2) + \log_{e}(3) \]

Using the property of logarithms that states \( \log_{e}(a) + \log_{e}(b) = \log_{e}(ab) \):

\[ \log_{e}(2) + \log_{e}(3) = \log_{e}(2 \cdot 3) = \log_{e}(6) \]

### Final Answer

Thus, the value of the definite integral is:

\[ \int_{1}^{4} \log_{e}[x] \, dx = \log_{e}(6) \]