Question

If the integral is of the type $\int R(x,\sqrt{a{x}^2+bx+c})$ then which of the following substitutions can be used?

• A. $\sqrt{a{x}^2+bx+c}=xt\pm \sqrt{t}$
• B. $\sqrt{a{x}^2+bx+c}=\pm x\sqrt{a}+t$
• C. $\sqrt{a{x}^2+bx+c}=(x-\alpha )t$
• D. None

Answer 1

The correct option is B $\sqrt{a{x}^2+bx+c}=\pm x\sqrt{a}+t$

Euler substitution:
When $a>0$ we introduce a new variable $t$ by setting
$\sqrt{a{x}^2+bx+c}=x\sqrt{a}+t$
When $a<0$ we introduce a new variable $t$ by setting
$\sqrt{a{x}^2+bx+c}=-x\sqrt{a}+t$
Hence, $\sqrt{a{x}^2+bx+c}=\pm x\sqrt{a}+t$