bamajon1974
- 22
- 5
Does anyone know of a derivation or justification of Euler's substitution formulas for evaluating irrational expressions? In other words, to evaluate integrals of the form:
\int R(x,\sqrt{ax^2+bx+c})
You can use Euler's substitutions:
1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0
2. \sqrt{ax^2+bx+c} = tx \pm \sqrt{c}, c>0
3. \sqrt{ax^2+bx+c} = \sqrt{a(x-{x_{1}})(x-x_{2})}=t(x-x_{1})=t(x-x_{2})
if x_{1}, x_{2} are real.
I understand how to mechanically evaluate the integrals but cannot find any justification or derivation of the substitution formulas, especially when to use, for example, t+x as opposed to t-x in the first instance.Can anyone help please? Thanks!
\int R(x,\sqrt{ax^2+bx+c})
You can use Euler's substitutions:
1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0
2. \sqrt{ax^2+bx+c} = tx \pm \sqrt{c}, c>0
3. \sqrt{ax^2+bx+c} = \sqrt{a(x-{x_{1}})(x-x_{2})}=t(x-x_{1})=t(x-x_{2})
if x_{1}, x_{2} are real.
I understand how to mechanically evaluate the integrals but cannot find any justification or derivation of the substitution formulas, especially when to use, for example, t+x as opposed to t-x in the first instance.Can anyone help please? Thanks!