I am trying to learn how to work with the floor function, and I would really appreciate if someone can explain me the reason behind:
$ \int_{0}^{\infty} \lfloor x \rfloor x dx= \sum_{0}^{\infty}n\int_{n}^{n+1}xdx $
It sort of makes sense in my head, but I am confused - how can I simply change the integral in that way? since when (kidding) separating a product of two functions is even legal?
I guess it has something to do with the fact that $\lfloor x \rfloor$ is constant for every $n\leq x \leq n+1$ and n is natural, but I can't justify that with real rigorous claims.
Thanks.