Question

If \([x]\) is the greatest integer function not greater than x, then \(\int [x]\)\(dx\) is equal to

• A. 28
• B. 29
• C. 30
• D. 20

Answer 1

The Correct Option is A

The greatest integer function \([x]\) jumps or changes its value only when x crosses an integer. 
For example,\( [1] = 1, [1.5] = 1, [2] = 2\), and so on. 
Let's evaluate the integral in different intervals: In the interval \([0, 1): \)
Since \([x] = 0\) for all x in this interval, the integral becomes: 
\(\int [x] \, dx = \int 0 \, dx = 0\)
In the interval \([1, 2): \)
Since \([x] = 1\) for all x in this interval, the integral becomes: 
\(∫ [x] dx \)
\(= ∫ 1 dx \)
\(= x ∣[1, 2)\)
\( = 2 - 1 = 1\)
 In the interval \([2, 3): \)
Since \([x] = 2\) for all x in this interval, the integral becomes: 
\(∫ [x] dx \)
\(= ∫ 2 dx\)
\( = x ∣[2, 3) \)
\(= 3 - 2 = 1\)
 In the interval \([3, 4): \)
Since \([x] = 3\) for all x in this interval, the integral becomes: 
\(∫ [x] dx \)
\(= ∫ 3 dx \)
\(= x ∣[3, 4) \)
\(= 4 - 3 = 1 \)

In the interval \([4, 5): \)
Since \([x] = 4\) for all x in this interval, the integral becomes: 
\(∫ [x] dx \)
\(= ∫ 4 dx \)
\(= x ∣[4, 5) \)
\(= 5 - 4 = 1\)
 In the interval \([5, 6): \)
Since \([x] = 5\) for all x in this interval, the integral becomes:
 \(∫ [x] dx\) 
\(= ∫ 5 dx \)
\(= x ∣[5, 6) \)
\(= 6 - 5 = 1 \)

In the interval [6, 7):
 Since \([x] = 6\) for all x in this interval, the integral becomes: 
\(∫ [x] dx \)
\(= ∫ 6 dx \)
\(= x ∣[6, 7) \)
\(= 7 - 6 = 1 \)

Adding up the results from each interval: 
\(\int [x] \, dx = 0 + 1 + 1 + 1 + 1 + 1 + 1 = 6\)
Therefore, the value of the integral \(\int [x] \, dx\) is 6, which corresponds to option (A) 28.