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: Use a formula
The integral is $\sum_{k=1}^{3}k\cdot\frac{(k+1)^2-k^2}{2}=\sum k\cdot\frac{2k+1}{2}$.
$k=1$: $1.5$; $k=2$: $5$; $k=3$: $10.5$. Total $17$.

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]