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Integral - an alternative to expanding the denominator?

  1. May 30, 2012 #1
    Integral -- an alternative to expanding the denominator?

    The following integral can be easily solved by expanding the denominator but I am wondering if there is a another way to solve it

    [itex]\int\frac{1}{(x+1)^7-x^7-1}dx[/itex]
     
  2. jcsd
  3. May 30, 2012 #2
    Re: Integral

    You could try doing integration by substitution, and then repeating it several (probably 7) times. But I'm sure you would either end up with something really ugly or encounter a problem pretty early on that stops you.
     
  4. May 30, 2012 #3
    Re: Integral



    Would you be so kind as to show us how this integral is "easily" solved expanding whatever? I think this is a rather

    horrible integral, expanding or not, and more than finding a way to make it more or less normal I'd love to see what's your way to solve it.

    Thanx

    DonAntonio
     
  5. May 30, 2012 #4
    Re: Integral

    By the binomial theorem we get
    [itex](x+1)^7-x^7-1=7(x^6+3x^5+5x^4+3x^2+x)[/itex]
    now if we factorize this term we also get
    [itex]x(x+1)(x^2+x+1)^2[/itex]
    then partial fraction completes the solution.
     
  6. May 30, 2012 #5
    Re: Integral


    Well, yes...but for this you must first (1) know how to factorize the polynomial (a quintic, since zero is obvious), perhaps by "guessing

    that -1 is a root, and (2) you must still make the partial fractions stuff, which seems far from being that easy, as [tex]\frac{1}{x(x+1)(x^2+x+1)^2}=\frac{A}{x}+\frac{B}{x+1}+\frac{Cx+D}{x^2+x+1}+\frac{Ex+F}{(x^2+x+1)^2}[/tex]
    A matter of taste, I guess...perhaps because I'm a theoretical mathematician I wouldn't dare call the above "easy", or perhaps I would

    but I'd add immediately "annoying and long" after that.

    DonAntonio
     
  7. May 30, 2012 #6
    Re: Integral

    Yes you are right, "easy" was not appropriate.
     
  8. May 31, 2012 #7
    Re: Integral -- an alternative to expanding the denominator?

    Hi !
    Who said "it's not easy" ?
    It takes less time to compute it that to type it. :redface:
     

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  9. May 31, 2012 #8
    Re: Integral -- an alternative to expanding the denominator?



    Really...?! Common, it's nice to show off but not with this petty things.

    DonAntonio
     
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