Definition:Euler Substitution/Second

Proof Technique
Euler's second 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 $c > 0$.

Euler's second substitution is the substitution:


 * $\ds \sqrt {a x^2 + b x + c} =: x t \pm \sqrt c$

Then:


 * $x = \dfrac {\pm 2 t \sqrt c - b} {a - t^2}$

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

Either the positive sign or negative sign can be used, according to what may work best.