Question

\(\int_{1}^{4} \log_e [x] dx\) equals

• A. \(\log_e 2\)
• B. \(\log_e 3\)
• C. \(\log_e 6\)
• D. None of the above

Hint:

For \(1 \le x<2\), \([x]=1\); for \(2 \le x<3\), \([x]=2\); for \(3 \le x<4\), \([x]=3\).

Answer 1

The Correct Option is C

To evaluate the integral \(\int_{1}^{4} \log_e [x] \, dx\), we begin by understanding the meaning of the function \([x]\), which denotes the greatest integer function (floor function). This function returns the greatest integer less than or equal to \(x\).

Thus, the range of the integer values of \([x]\) from 1 to 4 is: 

• \([x] = 1\) when \(1 \leq x < 2\)
• \([x] = 2\) when \(2 \leq x < 3\)
• \([x] = 3\) when \(3 \leq x < 4\)

We need to break the integral into parts where \([x]\) is constant:

1. For \(1 \leq x < 2\): \(\int_{1}^{2} \log_e [x] \, dx = \int_{1}^{2} \log_e 1 \, dx = \int_{1}^{2} 0 \, dx = 0\)
2. For \(2 \leq x < 3\): \(\int_{2}^{3} \log_e [x] \, dx = \int_{2}^{3} \log_e 2 \, dx = \log_e 2 \cdot (3 - 2) = \log_e 2\)
3. For \(3 \leq x < 4\): \(\int_{3}^{4} \log_e [x] \, dx = \int_{3}^{4} \log_e 3 \, dx = \log_e 3 \cdot (4 - 3) = \log_e 3\)

By summing these results, we find:

\(\int_{1}^{4} \log_e [x] \, dx = 0 + \log_e 2 + \log_e 3 = \log_e 2 + \log_e 3\)

Applying the logarithmic property \(\log_e a + \log_e b = \log_e (a \cdot b)\), we get:

\(\log_e 2 + \log_e 3 = \log_e (2 \cdot 3) = \log_e 6\)

Thus, the correct answer is \(\log_e 6\).