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Is it possible to solve this integral ?

  1. Jul 7, 2011 #1
    Is it possible to solve this integral ??

    1. The problem statement, all variables and given/known data
    Is it possible to solve this integral equation analytically. Actually I wanted to curve fit some data in matlab using a equation of which this is a part. I tried various techniques all in vain. PLease help..


    2. Relevant equations
    int((x^4)*(e^2)/((e^x)-1)^2)

    3. The attempt at a solution
     
  2. jcsd
  3. Jul 7, 2011 #2

    Ray Vickson

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    Re: Is it possible to solve this integral ??

    It involves "non-elementary" functions. Here is what I get using Maple:
    f:=x^4/(exp(x)-1)^2:
    J:=int(f,x);
    Maple's answer is
    J:= -x^4/(exp(x)-1) + x^5 /5 + (12x^2 - 4x^3) polylog(2,exp(x)) +
    (12x^2 - 24) polylog(3,exp(x)) + (24 - 24x) polylog(4,exp(x)) +
    24 polylog(5,exp(x)) - x^4 + (4x^3 - x^4) log(1-exp(x))
    Here, Maple uses the non-elementary function
    polylog(a,z) = sum_{n=1..infinity} z^n / n^a .


    RGV
     
  4. Jul 8, 2011 #3
    Re: Is it possible to solve this integral ??

    Thanks for replying, I read the article in non elementary intergrals, however there isnt satisfactory explanation given. I'm doing undergraduate and have never heard of non elementary integrals. Is that mean that you cant solve these integrals analytically without computers and how this is different from numerical integration. Even in numerical integration we can find integration and then fit a polynomial or any other taylor series. So we wont call that a integral.

    BTW I tried to compute this in matlab symbolic toolkit, it says its indeterminable. How can mathematica solve it. can mathematica solve integrals better than matlab???

     
  5. Jul 8, 2011 #4

    Ray Vickson

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    Re: Is it possible to solve this integral ??

    I don't have Mathematica (I use Maple instead), but since the integral is not elementary, Mathematica will not be able to do anything very different. Perhaps it would express the answer using non-elementary functions different from Maple's polylog, but the result will still be in terms of non-elementary functions.

    Non-elementary integrals are just integrals for which we have no finite, closed-form formula. They appear all over the place, in many types of problems. For example, the integrals of sin(x)/x or exp(x)/x are non-elementary; the integral of exp(-x^2) is non-elementary; the integral of sqrt((1-x^2)*(1-k*x^2)) is non-elementary, etc. Nevertheless, whether or not an integral is elementary, we still need methods of computing it numerically, and many ways exist of doing that (such as good approximations, series expansions, or straight numerical methods, etc.) Some scientific calculators have buttons that give the integral of exp(-x^2), for example, and all spreadsheets have similar capabilities. Programs like Maple or Mathematica can compute a wide variety of such functions, and are so easy to use that the distinction between "elementary" and "non-elementary" almost disappears.

    RGV
     
  6. Jul 8, 2011 #5
    Re: Is it possible to solve this integral ??

    Thanks a lot, for your help
     
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