Question

Let \[ f(x)=\int_{1}^{4}\log[x]\ dx, \] where \([x]\) denotes the greatest integer function. Then the value of \(f(x)\) is:

• A. \(\log 2\)
• B. \(\log 3\)
• C. \(\log 5\)
• D. \(\log 6\)

Hint:

Whenever the greatest integer function appears inside an integral, always divide the interval at every integer point. This converts the function into constant values over smaller intervals, making the integration very simple.

Answer 1

The Correct Option is D

Concept:
The greatest integer function \([x]\) gives the greatest integer less than or equal to \(x\). It behaves like a step function: \[ [x]=n \quad \text{for} \quad n\le x<n+1 \] Whenever a greatest integer function appears inside an integral, the interval must be broken into smaller intervals where the value of \([x]\) remains constant.

Step 1: Split the interval at integer points.
The integral is: \[ f(x)=\int_{1}^{4}\log[x]\ dx \] The integers between \(1\) and \(4\) are: \[ 2,\ 3 \] Therefore: \[ f(x) = \int_{1}^{2}\log[x]\ dx + \int_{2}^{3}\log[x]\ dx + \int_{3}^{4}\log[x]\ dx \]

Step 2: Evaluate the greatest integer function in each interval.
For: \[ 1\le x<2 \] we have: \[ [x]=1 \] Hence: \[ \log[x]=\log1=0 \] For: \[ 2\le x<3 \] we have: \[ [x]=2 \] Hence: \[ \log[x]=\log2 \] For: \[ 3\le x<4 \] we have: \[ [x]=3 \] Hence: \[ \log[x]=\log3 \] Thus: \[ f(x) = \int_{1}^{2}0\ dx + \int_{2}^{3}\log2\ dx + \int_{3}^{4}\log3\ dx \]

Step 3: Evaluate each integral separately.
First integral: \[ \int_{1}^{2}0\ dx=0 \] Second integral: \[ \int_{2}^{3}\log2\ dx = \log2[x]_{2}^{3} \] \[ =\log2(3-2) \] \[ =\log2 \] Third integral: \[ \int_{3}^{4}\log3\ dx = \log3[x]_{3}^{4} \] \[ =\log3(4-3) \] \[ =\log3 \] Therefore: \[ f(x)=\log2+\log3 \]

Step 4: Use logarithmic identity.
Using: \[ \log m+\log n=\log(mn) \] we get: \[ f(x)=\log(2\times3) \] Hence: \[ f(x)=\log6 \] Therefore: \[ \boxed{f(x)=\log6} \]

Step 5: Verification with the options.
Comparing with the given options: \[ (A)\ \log2 \] \[ (B)\ \log3 \] \[ (C)\ \log5 \] \[ (D)\ \log6 \] the correct answer is: \[ \boxed{(D)\ \log6} \]