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.

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