How to evaluate $\int_{0} ^{2}\frac{(x-⌊x⌋)dx}{2x+1-⌊x⌋}$

Question

I want to evaluate $\int_{0} ^{2}\frac{(x-⌊x⌋)dx}{2x+1-⌊x⌋}$. How can I do this? Can it be done just with the graph of the function? We have $[x] =x-\{x\}$ and after this the graph would follow. But without that, how can it be solved?

Is there a general way to approach such functions?

Answer 1

$x-\lfloor x\rfloor=n+\{x\}$ where $n=\lfloor x\rfloor$

$2x+1-\lfloor x\rfloor=2n+\{x\}+1$

so we can split up your integral into two parts: $$f(x)=\frac{x-\lfloor x\rfloor}{2x+1-\lfloor x\rfloor}=\begin{cases}\frac{y}{y+1} & 0\le x<1\\\frac{1+y}{3+y}&1\le x<2\end{cases}$$ where $y\in[0,1)$