Integration using Euler's formula: Difference between revisions

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{{calculus|expanded=integral}}

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

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/}}</ref>

==Euler's formula==

Euler's formula states that <ref>[http{{Cite web|last=Weisstein|first=Eric W.|title=Euler Formula|url=https://mathworld.wolfram.com/EulerFormula.html Weisstein, Eric W|access-date=2021-03-17|website=mathworld.(June 14 2017)]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