Atal's Integrals

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Given a number $r$, define a function $f_r$ on $[0,1]$ as follows: $$f_r(x) =\begin{cases} x^r \left( \frac{1}{x}-n \right) & \text{if } n \in \mathbb{N} \text{ and } \frac{1}{n+1} \le x \leq \frac{1}{n}, \\ 0 & \text{if } x = 0. \end{cases}$$ If $r > 1$, $f_r$ has bounded variation and so we can consider Stieltjes integrals with respect to $f_r$.
The following links contain pages with the values of the sum $$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_r(t)$$ as $N$ and $r$ vary.

For integer $r$:

$\underline{r=2}$
$\underline{r=3}$
$\underline{r=4}$
$\underline{r=5}$
$\underline{r=6}$
$\underline{r=7}$
$\underline{r=8}$
$\underline{r=9}$
$\underline{r=10}$
$\underline{r=11}$

For half-integer $r$:

$\underline{r=\frac{3}{2}}$
$\underline{r=\frac{5}{2}}$
$\underline{r=\frac{7}{2}}$
$\underline{r=\frac{9}{2}}$
$\underline{r=\frac{11}{2}}$

Atal's Integrals

For integer \(r\):

For half-integer \(r\):