Indefinite integral problem: $\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$

Question

$ I =\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$

my attempt: $I=\int_1^n \frac{(x - [x])^{[x]}{[x]}}\implies\sum_{i=1}^n\int_r^{r+1} \frac{(x-r)^r}{r}dx $

Now by integration I got,

$\implies \left.\frac{(x-r)^{r+1}}{r(r+1)}\right|_{r}^{r+1} +\frac{1}{n(n+1)} \implies \sum_{r=1}^n \frac{1}{r(r+1)} $

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But now I'm stuck at summation, any hints?

Answer 1

Here's a standard trick which shows you that the sum telescopes and is thus easy to evaluate: $$\frac{1}{r(r+1)}=\frac{1}r-\frac{1}{r+1}$$