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.

Also see

• Results about Euler substitutions can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• This article incorporates material from Euler’s substitutions for integration on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.