Euler substitution

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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[edit]

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[edit]

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[edit]

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[edit]

Examples for Euler's first substitution[edit]


In the integral we can use the first substitution and set , thus

Accordingly, we obtain:

The cases give the formulas


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


Examples for Euler's second substitution[edit]

In the integral

we can use the second substitution and set . Thus

Accordingly, we obtain:

Examples for Euler's third substitution[edit]

To evaluate

we can use the third substitution and set . Thus


As we can see this is a rational function which can be solved using partial fractions.


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[edit]


  1. ^ 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.
  2. ^ Zwillinger, Daniel. The Handbook of Integration. 1992: Jones and Bartlett. pp. 145–146. ISBN 978-0867202939.CS1 maint: location (link)

This article incorporates material from Eulers Substitutions For Integration on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.