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Topic: integral for fun
Replies: 25   Last Post: Mar 6, 2014 4:10 PM

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Axel Vogt

Posts: 1,038
Registered: 5/5/07
Re: integral for fun
Posted: Feb 28, 2014 4:00 PM
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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).




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