Definition:Euler Substitution/Third

Proof Technique
Euler's third substitution is a technique for evaluating primitives of the form:
 * $\ds \map R {x, \sqrt {a x^2 + b x + c} } \rd x$

where $R$ is a rational function of $x$ and $\sqrt {a x^2 + b x + c}$.

Let $a x^2 + b x + c$ have real roots $\alpha$ and $\beta$.

Euler's third substitution is the substitution:


 * $\ds \sqrt {a x^2 + b x + c} = \sqrt {a \paren {x - \alpha} \paren {x - \beta} } = \paren {x - \alpha} t$

Then:


 * $x = \dfrac {a \beta - \alpha t^2} {a - t^2}$

and hence $\d x$ is expressible as a rational function of $x$.

Also known as
Euler's third substitution is also known as an Euler substitution of the third kind.

Also see

 * Lemmata for Euler's Third Substitution