Method of integration for rational functions
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- Mean value theorem
- Inverse function theorem
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Differential Definitions |
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- Second derivative
- Implicit differentiation
- Logarithmic differentiation
- Related rates
- Taylor's theorem
| Rules and identities |
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| Definitions |
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| Integration by |
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Euler substitution is a method for evaluating integrals of the form
where is a rational function of and . In such cases, the integrand can be changed to a rational function by using the substitutions of Euler.[1]
Euler's first substitution
The first substitution of Euler is used when . We substitute
and solve the resulting expression for
. We have that
and that the
term is expressible rationally in
.
In this substitution, either the positive sign or the negative sign can be chosen.
Euler's second substitution
If , we take
We solve for
similarly as above and find
Again, either the positive or the negative sign can be chosen.
Euler's third substitution
If the polynomial has real roots and , we may choose . This yields and as in the preceding cases, we can express the entire integrand rationally in .
Worked examples
Examples for Euler's first substitution
One
In the integral we can use the first substitution and set , thus
Accordingly, we obtain:
The cases give the formulas
Two
For finding the value of
we find
using the first substitution of Euler,
. Squaring both sides of the equation gives us
, from which the
terms will cancel out. Solving for
yields
From there, we find that the differentials and are related by
Hence,
Examples for Euler's second substitution
In the integral
we can use the second substitution and set
. Thus
and
Accordingly, we obtain:
Examples for Euler's third substitution
To evaluate
we can use the third substitution and set
. Thus
and
Next,
As we can see this is a rational function which can be solved using partial fractions.
Generalizations
The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral , the substitution can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
The substitutions of Euler can be generalized to a larger class of functions. Consider integrals of the form
where
and
are rational functions of
and
. This integral can be transformed by the substitution
into another integral
where
and
are now simply rational functions of
. In principle,
factorization and
partial fraction decomposition can be employed to break the integral down into simple terms, which can be integrated analytically through use of the
dilogarithm function.
[2] See also
- Mathematics portal
References
- ^ N. Piskunov, Diferentsiaal- ja integraalarvutus körgematele tehnilistele öppeasutustele. Viies, taiendatud trukk. Kirjastus Valgus, Tallinn (1965). Note: Euler substitutions can be found in most Russian calculus textbooks.
- ^ Zwillinger, Daniel. The Handbook of Integration. Jones and Bartlett. pp. 145–146. ISBN 978-0867202939.
This article incorporates material from Eulers Substitutions For Integration on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.