Question about the proof of Theorem 3.1 (Page 54) of the book Introduction to Analytic Number Theory by Apostol

Question

Why $\displaystyle\int_{n-1}^{n}[t]f'(t)dt=\int_{n-1}^{n}(n-1)f'(t)dt$? IMO, since $t\leq n$, $[t]$ could be $n$. How to explain this? Thanks in advance!

This is used in the proof of Euler's summation formula (Theorem 3.1) in Apostol's Introduction to Analytic Number Theory, page 54.

Answer 1

I assume $[t]$ denotes the integer part.

Then $[t]=n$ only when $t=n$. If two functions only differ at finitely many points, their integrals are equal.

Answer 2

$[t]=n$ at only one point, and no one point affects an integral.