Integration using Euler's formula: Difference between revisions

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

{{unreferenced|date=October 2016}}

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

In [[integral calculus]], [[complex number]]s and [[Euler's formula]] may be used to evaluate [[integral]]s involving [[trigonometric functions]]. Using Euler's formula, any trigonometric function may be written in terms of {{math|''e''^''ix''}} and {{math|''e''^−''ix''}}, 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 expression]] involving trigonometric functions.

{{calculus|expanded=integral}}

In [[integral calculus]], [[complexEuler's numberformula]]s andfor [[Euler'scomplex formulanumber]]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''<sup>''^{ix''}</supmath>}} and {{<math|''>e''<sup>−''^{-ix''}</supmath>}}, 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|journal=American Review of Mathematics and Statistics|date=2019 |publisher=American Research Institute for Policy Development|volume=7|pages=1–2|doi=10.15640/arms.v7n2a1|doi-broken-date=12 July 2025 |issn=2374-2348|eissn=2374-2356|doi-access=free|url=https://arms.thebrpi.org/vol-7-no-2-december-2019-abstract-1-arms |hdl=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>

Euler's formula states that

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

Substituting {{<math|−''>-x''}}</math> for {{<math|''>x''}}</math> gives the equation

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

because cosine is an even function and sine is odd. These two equations can be solved for the sine and cosine: to give

:<math>\cos x = \frac{e^{ix} + e^{-ix}}{2}\quad\text{and}\quad\sin x = \frac{e^{ix}-e^{-ix}}{2i}.</math>

==Simple exampleExamples==

=== First example ===

Consider the integral

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

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

:<math>\begin{align}

\int \cos^2 x \, dx \,&=\, \int \left(\frac{e^{ix}+e^{-ix}}{2}\right)^2 dx \\[6pt]

&=\, \frac14\int \left( e^{2ix} + 2 + e^{-2ix} \right) dx

\end{align}</math>

At this point, it would be possible to change back to real numbers using the formula {{math|''e''^2''ix'' + ''e''^−2''ix'' {{=}} 2 cos 2''x''}}. Alternatively, we can integrate the complex exponentials and not change back to trigonometric functions until the end:

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\end{align}</math>

===Second example===

Consider the integral

:<math>\int \sin^2 x \cos 4x \, dx.</math>

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&= -\frac18\int \left(e^{6ix} - 2e^{4ix} + e^{2ix} + e^{-2ix} - 2e^{-4ix} + e^{-6ix}\right) dx.

\end{align}</math>

At this point we can either integrate directly, or we can first change the integrand to {{math|2 cos 6''x'' − 24 cos 4''x'' + 2 cos 2''x''}} and continue from there.

Either method gives

:<math>\int \sin^2 x \cos 4x \, dx = -\frac{1}{24} \sin 6x + \frac18\sin 4x - \frac18 \sin 2x + C.</math>

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Using Euler's identity, this integral becomes

:<math>\frac12 \int \frac{6 + e^{2ix} + e^{-2ix} }{e^{ix} + e^{-ix} + e^{3ix} + e^{-3ix}} \, dx.</math>

If we now make the [[integration by substitution|substitution]] {{<math|''>u'' {{=}} ''e''<sup>''^{ix''}</supmath>}}, the result is the integral of a [[rational function]]:

:<math>-\frac{i}{2}\int \frac{1+6u^2 + u^4}{1 + u^2 + u^4 + u^6}\,du.</math>

One may proceed using [[partial fraction decomposition]].

Any [[rational function]] is integrable (using, for example, [[partial fractions in integration|partial fractions]]), and therefore any fraction involving trigonometric functions may be integrated as well.

==See also==

{{Portal|Mathematics}}

* [[Trigonometric substitution]]

* [[Weierstrass substitution]]

* [[Euler substitution]]

==References==

{{Reflist}}

{{Integrals}}

[[Category:Integral calculus]]

[[Category:Theorems in mathematical analysis]]

[[Category:Theorems in calculus]]