Can someone please explain why this is the first move when solving this integral. The integral which I am solving is $$\int_0^c\sqrt{c^2-x^2}~\mathrm dx$$ and the first thing which they suggest doing is setting $x=(\sqrt{a})/(\sqrt{b}) \sin u$. I understand everything after this but I do not comprehend where and why this step came to be. 

Maybe it is better to divide the procedure into two steps.
First by letting $X=x/c$ (here we assume $c>0$) we obtain
$$\int_0^c\sqrt{c^2-x^2}\,dx=c^2\int_0^1\sqrt{1-X^2}\,dX$$
Now it seems quite natural to use the substitution $X=\sin u$.
