# 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 $$r$$:

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