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Non integrable functions

  1. Dec 20, 2004 #1
    hai
    i heard that the function x^x doesnt have any indefinite integral and hence one cant find definite integral by normal methods .. So one has to go for numerical methods , i havent tried this ...
    just curious to know if there exist more fuctions in the same class ...

    regards
    mahesh :smile:
     
  2. jcsd
  3. Dec 20, 2004 #2
    Well, the integral "exists" of course, but it's not the expressible in elementary functions.

    There are many such integrals of this nature.
     
  4. Dec 20, 2004 #3
    Thanx wolfe, can u tell me where i can find such functions

    Mahesh
     
  5. Dec 20, 2004 #4

    dextercioby

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    Yap,just about elliptic integrals.Basically most of the type [itex] \sqrt{P(x)} [/itex],where P(x) is a polynomial with real coeffcients of degree larger of equal with 3,get the "chance" of not having a "cute" antiderivative.Mathematicians invented the famous syntagma "nonelementary function",referring to this sort of functions which come up when searching for antiderivatives.They couldn't come up with a decent definition for this "nonelementary". :tongue2:

    Anyway,when you spot something wrong,i.e.u can't find an antiderivative,try for other tools.Numerical analysis works,but only in the case on definite integral,where the result is a number.Sometimes,u can expand the integrand in Taylor series (though the ray may be small) or express it terms on tabulated "nonelementary functions".The books by Abramowitz & Stegun and Gradsteyn & Rytzhik may turn out to be handy.

    Daniel.

    PS.If the antiderivatives exist,but cannot be expressed in terms of "elementary" functions,then the function which makes up the integrand is integrable.
     
    Last edited: Dec 20, 2004
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