Euler substitution
E480876
Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Euler substitution canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4927386 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Euler substitution Context triple: [Weierstrass substitution, relatedTo, Euler substitution]
-
A.
Weierstrass substitution
The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
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B.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
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C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
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D.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
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E.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Euler substitution Target entity description: Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
-
A.
Weierstrass substitution
The Weierstrass substitution is a trigonometric substitution technique that transforms integrals involving sine and cosine into rational functions, simplifying their evaluation.
-
B.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
-
C.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
D.
Euler’s theorem
Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
-
E.
Euler’s method of rearranging absolutely convergent series
Euler’s method of rearranging absolutely convergent series is a technique introduced by Leonhard Euler to systematically reorder and manipulate convergent infinite series in order to derive new identities and product expansions, such as those appearing in analytic number theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Euler substitution variant
ⓘ
Euler substitution variant ⓘ Euler substitution variant ⓘ calculus technique ⓘ integration technique ⓘ mathematical method ⓘ |
| advantage | provides a systematic method for certain irrational integrals ⓘ |
| appearsIn |
advanced calculus textbooks
ⓘ
integration technique chapters in mathematical analysis books ⓘ |
| appliesTo |
integrals of the form ∫ R(x, sqrt(a x^2 + b x + c)) dx
ⓘ
integrals with quadratic expressions under a square root ⓘ |
| basicIdea |
express x as a rational function of t using the roots of the quadratic
ⓘ
set sqrt(a x^2 + b x + c) equal to sqrt(-a) x + t or similar expression ⓘ set sqrt(a x^2 + b x + c) equal to t - sqrt(a) x or similar linear expression in x and t ⓘ |
| category | techniques of integration ⓘ |
| complexity | can lead to lengthy algebraic manipulations ⓘ |
| effect |
eliminates the square root from the integrand
ⓘ
reduces the integral to one involving only rational functions ⓘ |
| field | integral calculus ⓘ |
| goal | to obtain an integral of a rational function in the new variable ⓘ |
| hasVariant |
first Euler substitution
ⓘ
second Euler substitution ⓘ third Euler substitution ⓘ |
| historicalAttribution | introduced by Leonhard Euler ⓘ |
| limitation |
mainly useful for quadratic expressions under the square root
ⓘ
may be less convenient than trigonometric substitution in some cases ⓘ |
| mathematicalDomain |
classical analysis
ⓘ
real analysis ⓘ |
| namedAfter | Leonhard Euler ⓘ |
| purpose |
to simplify integrals involving square roots of quadratic expressions
ⓘ
to transform irrational integrals into rational integrals ⓘ |
| relatedTo |
rational substitution
ⓘ
trigonometric substitution ⓘ |
| requires |
ability to perform partial fraction decomposition
ⓘ
algebraic manipulation of quadratic polynomials ⓘ |
| resultType | rational function of the new variable ⓘ |
| typicalCondition |
a < 0 in sqrt(a x^2 + b x + c)
ⓘ
a > 0 in sqrt(a x^2 + b x + c) ⓘ quadratic has real roots ⓘ |
| typicalIntegrandForm |
sqrt(a x^2 + b x + c)
ⓘ
sqrt(a x^2 + b x + c) in numerator or denominator ⓘ |
| typicalOutcome | integral expressible in terms of logarithms and arctangents ⓘ |
| usedBy |
engineers
ⓘ
mathematicians ⓘ physicists ⓘ |
| usedIn |
manual integration techniques taught in calculus courses
ⓘ
symbolic integration ⓘ |
| uses | change of variables ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Euler substitution Description of subject: Euler substitution is a classical technique in integral calculus that simplifies integrals involving square roots of quadratic expressions by transforming them into rational functions through a specific change of variables.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.