Definition:Euler Substitution/First

Proof Technique
Euler's first 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 $a > 0$.

Euler's first substitution is the substitution:


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

Then:


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

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 first substitution is also known as an Euler substitution of the first kind.