Thursday, February 8, 2018

definite integrals - Compute intinfty0fracsqrtxsin(x)1+x2dx



How does one compute the following integral?
0xsin(x)1+x2dx







I have tried extending x to the complex plane then evaluating the following contour integral
Cxeix1+x2dx
with the contour C running along the whole real axis and then upper semicircle. I obtain
0x(cos(x)+sin(x))1+x2dx=πe2
but not the original integral.


Answer



For information :




0xsin(yx)1+x2dx is the Fourier Sine Transform of x1+x2 In the Harry Bateman's Tables of Integral Transforms an even more general formula can be seen on page 71, Eq.28 , the Fourier Sine Transform of x2ν(x2+a2)μ1 :
12a2ν2μΓ(1+ν)Γ(μν)Γ(μ+1)y1F2(ν+1;ν+1μ,3/2;a2y2/4)+4νμ1πΓ(νμ)Γ(μν+3/2)y2μ2ν+11F2(μ+1;μν+3/2,μν+1;a2y2/4)
Withy=1;a=1;ν=1/4;μ=00xsin(x)1+x2dx=
=12Γ(5/4)Γ(1/4)1F2(5/4;5/4,3/2;1/4)+43/4πΓ(5/4)Γ(5/4)1F2(1;5/4,3/4;1/4)
The Generalized Hypergeometric 1F2 function (don't confuse with the well-known 2F1) reduces to functions of lower level in the particular cases :



1F2(5/4;5/4,3/2;1/4)=sinh(1)




1F2(1;5/4,3/4;1/4)=π4e(e2erfi(1)+erf(1))



and after simplification :
0xsin(x)1+x2dx=π2sinh(1)+π22e(e2erfi(1)+erf(1))
0xsin(x)1+x2dx=π22e(e2erfi(1)+erf(1)+1e2)
0xsin(x)1+x2dx=π22e(e2erfc(1)+erf(1)+1)
For the Hypergeometric 1F2 function, see : http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F2/02/



About the functions erf, erfi, erfc, see : http://mathworld.wolfram.com/Erf.html , http://mathworld.wolfram.com/Erfi.html , http://mathworld.wolfram.com/Erfc.html


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