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.
Observed surface forms (3)
| Surface form | Occurrences |
|---|---|
| first Euler substitution | 0 |
| second Euler substitution | 0 |
| third Euler substitution | 0 |
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 NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.