
Re: integral for fun
Posted:
Feb 28, 2014 4:00 PM


On 28.02.2014 18:55, Axel Vogt wrote: > On 28.02.2014 15:32, Jonas Matuzas wrote: >> I wrote integral in Mathematica: >> Integrate[Cos[x]^4/(1 + x^8), {x, \[Infinity], \[Infinity]}] ... >> I am a bit lazy but I think this integral can by taken with pencil and paper using residual theory. Hm >> I find publication is talking about similar problem , and writing that in this case residual theory is not complete..: >> http://www.csm.ornl.gov/~bowman/fjts312.pdf ... > val := proc() ... > end proc
I looked in the linked paper, partially. Some  and your task  may be seen via Fourier. Write cos(x)^4 = 1/8*cos(4*x)+1/2*cos(2*x)+3/8 ("expand" in Mpl)
The constant is easy, for any CAS.
The other can be seen as real parts of a Fourier transform in t=4 and t=2. Or as 2*Cosinus Fourier, using symmetry in x=0. To be scaled in both cases.
Maple finds that, and MMA certainly as well.
For the task in the intro of the paper cos(x) / product of quadrics note that it can be written as sums of quadrics (by partial fractions) and one achieves Int( cos(x)/(a^2+x^2), x=infinity..infinity) / const and summing such terms.
But for each the above Fourier approach gives a 'compact' answer (and only looks complicated if feeding specific values).
That way one also can treat the constant denominator (but has to use limits).

