Question

If \([\cdot ]\) denotes the greatest integer function, then \(\int _1^4x[x]dx\) is equal to

• A. \(34\)
• B. \(-17\)
• C. \(-34\)
• D. \(17\)

Hint:

Split the interval at the integers where $[x]$ jumps.

Answer 1

The Correct Option is D

Step 1: Split
On \([1,2)\), \([x]=1\); on \([2,3)\), \([x]=2\); on \([3,4)\), \([x]=3\).

Step 2: Integrate each part
\[ \int_1^2x\,dx=\frac{4-1}{2}=\frac32 \]
\[ \int_2^32x\,dx=9-4=5 \]
\[ \int_3^43x\,dx=\frac{3(16-9)}{2}=\frac{21}{2} \]

Step 3: Add
\(\frac32+5+\frac{21}{2}=17\).

Final Answer:
The value is \(17\), option (D). \[ \boxed{\text{(D) }17} \]