Editing Dirichlet integral (section)

{{Use American English|date = January 2019}}
{{Short description|Integral of sin(x)/x from 0 to infinity}}
{{Distinguish|Dirichlet energy}} 
[[File:Dirichlet 3.jpeg|thumb|[[Peter Gustav Lejeune Dirichlet]]]]
{{calculus}}
In [[mathematics]], there are several [[integral]]s known as the '''Dirichlet integral''', after the German mathematician [[Peter Gustav Lejeune Dirichlet]], one of which is the [[improper integral]] of the [[sinc function]] over the positive real number line.

<math display="block">\int_0^\infty \frac{\sin x}{x} \,dx = \frac{\pi}{2}.</math>

This integral is not [[absolutely convergent]], meaning <math>\left| \frac{\sin x}{x} \right|</math> has infinite Lebesgue or Riemann improper integral over the positive real line, so the sinc function is not [[Lebesgue integrable]] over the positive real line. The sinc function is, however, integrable in the sense of the improper [[Riemann integral]] or the generalized Riemann or [[Henstock–Kurzweil integral]].<ref>{{cite journal |last=Bartle |first=Robert G. |author-link=Robert G. Bartle |date=10 June 1996 |title=Return to the Riemann Integral |url=http://math.tut.fi/courses/73129/Bartle.pdf |journal=The American Mathematical Monthly |volume=103 |issue=8 |pages=625–632 |doi=10.2307/2974874 |jstor=2974874 |access-date=10 June 2017 |archive-date=18 November 2017 |archive-url=https://web.archive.org/web/20171118184849/http://math.tut.fi/courses/73129/Bartle.pdf |url-status=dead }}</ref><ref>{{Cite book|last=Bartle|first=Robert G.|title=Introduction to Real Analysis|url=https://archive.org/details/introductiontore00bart_903|url-access=limited|last2=Sherbert|first2=Donald R.|publisher=John Wiley & Sons|year=2011|isbn=978-0-471-43331-6|pages=[https://archive.org/details/introductiontore00bart_903/page/n325 311]|chapter=Chapter 10: The Generalized Riemann Integral}}</ref> This can be seen by using [[Dirichlet%27s_test#Improper_integrals |Dirichlet's test for improper integrals]].  

It is a good illustration of special techniques for evaluating definite integrals, particularly when it is not useful to directly apply the [[fundamental theorem of calculus]] due to the lack of an elementary [[antiderivative]] for the integrand, as the [[sine integral]], an antiderivative of the sinc function, is not an [[elementary function]]. In this case, the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel. But since the integrand is an even function, the domain of integration can be extended to the negative real number line as well.