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

Answer: (A)

28

Answer 2

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.