**Question:** How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$?

$$\int_{-1}^1 \left(\sqrt{r^2-x^2}\right)^{d-1} dx\le c\cdot \frac{r^d}{\sqrt{d-1}}$$

(*Furthermore, is it also possible to find an upper bound for the minimum value of $c\in [1,\infty)$ such that the above inequality holds for all $d>1$ and all $r\in\left[1,\sqrt{d}\right]$?*)