Limit calculation with Riemann integral Help me calculating the limit with Riemann integral:
$$a_n=\frac{1}{n\sqrt{n^2+1}}+\frac{2}{n\sqrt{n^2+4}}+\frac{3}{n\sqrt{n^2+9}}+...+\frac{n-1}{n\sqrt{n^2+(n-1)^2}}$$

Hint: This can be written as
$$\lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^{n-1} \frac{r}{\sqrt{(n^2+r^2)}}$$
$$=\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n-1} \frac{\frac{r}{n}}{\sqrt{1+(\frac{r}{n})^2}}$$
$$=\int_0^1 \frac {xdx}{\sqrt{1+x^2}}$$
