Show that $\int_0^r \frac{\mathrm{d}t}{\sqrt{r^2 - t^2}} $ is independent of $r$

Question

I'm trying to show that the integral

$$\int_0^r \frac{1}{\sqrt{r^2 - t^2}}\mathrm{d}t$$

is independent of $r$, without using trigonometric functions (namely, $t=\cos s$ and such).

Can it be done?

Thanks!

Answer 1

Make a substitution $t = rs$, we get

$$\int_0^r \frac{1}{\sqrt{r^2 - t^2}}\mathrm{d}t = \int_0^1 \frac{1}{\sqrt{1 - s^2}}\mathrm{d}s$$