Definition:Euler Substitution/Third
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Proof Technique
Euler's third 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 $a x^2 + b x + c$ 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 known as
Euler's third substitution is also known as an Euler substitution of the third 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.