Question #40427

This question tests the student's ability to handle nested Greatest Integer Functions within a definite integral by rigorously applying inequality bounds.

The primary trap is the assumption that the inner GIF can be treated as a constant over its entire natural interval $[k^2, (k+1)^2)$, which would lead to evaluating $[\sqrt{x+k}]$ as simply $k$ for the whole sub-interval.

Students must recognize that $x+k$ crosses the next perfect square $(k+1)^2$ strictly inside this interval, specifically at $x = k^2+k+1$.

Failing to partition the integral at this hidden discontinuity collapses the inequality bounds, leading to incorrect summations.

A secondary trap involves off-by-one errors when evaluating the final truncated interval $[36, 45]$.