relative of cosine integral

For determining of the value of the improper integral

I(a):=0cosax2-cosaxxdx  (a>0),

related to the cosine integralDlmfDlmfDlmfMathworldPlanetmath, we think it as a functionMathworldPlanetmath of the parametre a which we denote by t.  Then we can take the Laplace transformDlmfMathworldPlanetmath (see the integration with respect to a parametre in the table of Laplace transforms):


Splitting the fractional expressions to fractions and integrating give

{I(t)} =1s0(1x-x3s2+x4-1x+xs2+x2)𝑑x

As seen in the of Laplace transforms, the gotten expression is the Laplace transform of  γ+lnt2=I(t)  (N.B.  {1}=1s), and thus we have the result

Title relative of cosine integral
Canonical name RelativeOfCosineIntegral
Date of creation 2013-03-22 18:44:35
Last modified on 2013-03-22 18:44:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Example
Classification msc 44A10
Related topic SubstitutionNotation
Related topic EulersConstant
Related topic RelativeOfExponentialIntegral
Related topic IntegrationOfLaplaceTransformWithRespectToParameter