As the title suggests, I want to know the motivation behind the Euler substitutions.How someone arrived at these substitutions.
To remove any doubts, by euler substitutions i mean -
If we have a rational function of $x$ and $\sqrt{ax^2+bx+c}$ ,
Case I. If $a>0$ , put $\sqrt{ax^2+bx+c}=t \pm x\sqrt{a}$
Case II. If $c>0$ , put $\sqrt{ax^2+bx+c}=tx \pm \sqrt{c}$
Case III. If the trinomial has real roots m,n, then put $\sqrt{ax^2+bx+c}= t(x-m)$
When considering $t=\sqrt{ax^2+bx+c}$ , you will find that $x$ in terms of $t$ is tedious.
But when considering e.g. $\sqrt{ax^2+bx+c}=t\pm x\sqrt{a}$ , $\sqrt{ax^2+bx+c}=tx\pm\sqrt{c}$ instead, you will find that $x$ in terms of $t$ are easier and give the rational functions.