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.

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.