Integrating floor functions without known limits Suppose we have$$\frac{\displaystyle\int_0^n{\lfloor x \rfloor}\,dx}{\displaystyle\int_0^n{\lbrace x \rbrace }\,dx}$$ $n \in I$
where $\lfloor \cdot\rfloor$ and $\lbrace \cdot\rbrace $ represent the floor function and fractional part function.
The usual method of splitting the function in intervals fails because the upper limit is not fixed. Is there any other way to approach this?
Edit:
I would also like to know a method to integrate $\displaystyle\int_0^n{\lbrace x \rbrace }\,dx$ without depending on floor function

You can still use the usual method:
$$\int_0^n \lfloor x\rfloor dx=\sum_{i=0}^{n-1}\int_i^{i+1}\lfloor x\rfloor dx=\sum_{i=0}^{n-1}i=\cdots$$
Can you finish from here?
