Integration using Euler's formula: Difference between revisions

Revision as of 20:07, 18 April 2021

Latest revision as of 04:50, 12 July 2025

(12 intermediate revisions by 9 users not shown)

Line 1:

{{refimprove|date=July 2019}}

{{calculus|expanded=integral}}

{{short description|Use of complex numbers to evaluate integrals}}

{{refimprovemore citations needed|date=July 2019}}

In [[integral calculus]], [[Euler's formula]] for [[complex number]]s may be used to evaluate [[integral]]s involving [[trigonometric functions]]. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely <math>e^{ix}</math> and <math>e^{-ix}</math> and then integrated. This technique is often simpler and faster than using [[trigonometric identities]] or [[integration by parts]], and is sufficiently powerful to integrate any [[rational fraction|rational expression]] involving trigonometric functions.<ref>{{Cite journal|last=Kilburn|first=Korey|title=Applying Euler’s Formula to Integrate|url=https://doi.org/10.15640/arms.v7n2a1|journal=American Review of Mathematics and Statistics|publisher=American Research Institute for Policy Development|volume=7|pages=1-2|doi=10.15640/arms.v7n2a1|issn=2374-2348|eissn=2374-2356|via=http://armsnet.info/|doi-access=free}}</ref>

{{calculus|expanded=integral}}

In [[integral calculus]], [[Euler's formula]] for [[complex number]]s may be used to evaluate [[integral]]s involving [[trigonometric functions]]. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely <math>e^{ix}</math> and <math>e^{-ix}</math> and then integrated. This technique is often simpler and faster than using [[trigonometric identities]] or [[integration by parts]], and is sufficiently powerful to integrate any [[rational fraction|rational expression]] involving trigonometric functions.<ref>{{Cite journal|last=Kilburn|first=Korey|title=Applying Euler’sEuler's Formula to Integrate|url=https://doi.org/10.15640/arms.v7n2a1|journal=American Review of Mathematics and Statistics|date=2019 |publisher=American Research Institute for Policy Development|volume=7|pages=1-21–2|doi=10.15640/arms.v7n2a1|doi-broken-date=12 July 2025 |issn=2374-2348|eissn=2374-2356|viadoi-access=httpfree|url=https://armsnetarms.infothebrpi.org/vol-7-no-2-december-2019-abstract-1-arms |doihdl=2158/1183208|hdl-access=free}}</ref>

==Euler's formula==

Euler's formula states that <ref>{{Cite web|last=Weisstein|first=Eric W.|title=Euler Formula|url=https://mathworld.wolfram.com/EulerFormula.html|access-date=2021-03-17|website=mathworld.wolfram.com|language=en}}</ref>

:<math>e^{ix} = \cos x + i\,\sin x.</math>

Substituting <math>-x</math> for <math>x</math> gives the equation

Line 17:

=== First example ===

Consider the integral

:<math>1|1+\int \cos^2 x \, dx .</math>

The standard approach to this integral is to use a [[half-angle formula]] to simplify the integrand. We can use Euler's identity instead:

:<math>\begin{align}

Line 77:

{{Reflist}}

{{Integrals}}

[[Category:Integral calculus]]

[[Category:Theorems in mathematical analysis]]

[[Category:Theorems in calculus]]