Definition:Euler Substitution

Proof Technique

Euler 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 $\sqrt {a x^2 + b x + c}$.

Euler's First Substitution

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.

Euler's Second Substitution

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.

Euler's Third Substitution

Let $a x^2 + b x + c$ have real roots $\alpha$ and $\beta$.

Euler's third substitution is the substitution:

$\ds \sqrt {a x^2 + b x + c} = \sqrt {a \paren {x - \alpha} \paren {x - \beta} } = \paren {x - \alpha} t$

Then:

$x = \dfrac {a \beta - \alpha t^2} {a - t^2}$

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

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.