Data Structures, Algorithms, & Applications in C++
Chapter 18, Exercise 25

We may show the equality by either integrating s loges by parts or differentiating both sides. Suppose we opt to differentiate both sides. The derivative of the left side is s loges, and that of the right side is s loges + s/2 - s/2 = s loges.

Using this result we get
integral (from 2 to m) s loges
= (m2logem)/2 - m2/4 - ((22loge2)/2 - 22/4)
= (m2logem)/2 - m2/4 - 2loge2 + 1
< (m2logem)/2 - m2/4.