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Elementary function for n > 0 is n=1

  1. Dec 20, 2004 #1
    [tex] \int x^n \cdot \sqrt{1-x^n} \ dx [/tex]
    It seems as the only time this is an elementary function for n > 0 is n=1 and n=2, can you prove / disprove this? n is an integer
     
  2. jcsd
  3. Dec 21, 2004 #2

    dextercioby

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    [tex] \int x^{n}\sqrt{1-x^{n}} dx=\int (-\frac{x}{n})(-nx^{n-1})\sqrt{1-x^{n}} dx=-\frac{2}{3}\frac{x}{n}(1-x^{n})^{\frac{3}{2}} +\frac{2}{3n}\int (1-x^{n})^{\frac{3}{2}} dx [/tex]

    The last integral can be solved immediately for "n=1" and through a sin/cos substitution for "n=2".As for "n>=3" (natural) it is impossible to solve analitically and express it through "elementary functions".

    Daniel.
     
  4. Dec 21, 2004 #3
    Can you prove that?
     
  5. Dec 21, 2004 #4

    dextercioby

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    I'm not a mathematician and i'm not claiming to be one.That assertion was purely based on my mathematical "flair" and on my past experience of solving integrals.For a proof or for a counterexample i'd advise you to consult a book which extensively covers integration in general and elliptic integrals in particular.


    Daniel.
     
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