Lemmata for Euler's Third Substitution

Theorem

Let Euler's third substitution be employed in order to evaluate a primitive 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 $\sqrt {a x^2 + b x + c}$.

Thus:

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

Then we have:

Lemma $1$

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

Lemma $2$

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

Lemma $3$

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

Lemma $4$

$t = \pm \sqrt {\dfrac {a \paren {x - \beta} } {x - \alpha} }$

Also see

• Definition:Euler's Third Substitution

• Results about Euler substitutions can be found here.