sine integral at infinity

The value of the improper integral (one of the Dirichlet integrals)


where Si means the sine integralDlmfDlmfDlmfMathworldPlanetmath ( functionMathworldPlanetmath, is most simply determined by using Laplace transformDlmfMathworldPlanetmath which may be aimed to the integrand (see integration of Laplace transform with respect to parameter).  Therefore the integrand must be equipped with an additional parametre t:


The obtained transform Ο€2β‹…1s corresponds (see the inverse Laplace transformation) to the function  t↦π2  because  ℒ⁒{1}=1s.  Thus we have the result

∫0∞sin⁑xx⁒𝑑x=Ο€2. (1)

Note 1.  Since  x↦sin⁑xx  or  x↦sinc⁑x  is an even function, the result (1) may be written also


see the sinc-function (

Note 2.  The result (1) may be easily generalised to

∫0∞sin⁑a⁒xxdx=Ο€2  (a>0) (2)

and to

∫0∞sin⁑a⁒xxdx=(sgna)Ο€2  (aβˆˆβ„). (3)
Title sine integral at infinity
Canonical name SineIntegralAtInfinity
Date of creation 2013-03-22 15:17:22
Last modified on 2013-03-22 15:17:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Derivation
Classification msc 44A10
Classification msc 30A99
Synonym limit of sine integral
Related topic SineIntegral
Related topic SincFunction
Related topic SubstitutionNotation
Related topic IncompleteGammaFunction
Related topic ExampleOfSummationByParts
Related topic SignumFunction