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Reduction formula

  1. Sep 23, 2009 #1
    1. The problem statement, all variables and given/known data

    In= (the integral) x(1-x^3)^ndx

    Prove that (3n +2)In = 3nIn-1 + x^2(1 - x)^n

    Hence find In in terms of n
    3. The attempt at a solution

    I tried integration by parts (by letting u be (1-x^3)^n and got stuck after this:
    In = 1/2x^2(1-x^3)^n + (the integral)(n(1-3x)^(n-1)(3x^2)(1/2x^2)dx)
    = 1/2(x^2(1-x^3)^n +3n(the integral)(1-x^3)^(n-1)x^4dx

    Would greatly appreciate any help and thanks in advance!
     
  2. jcsd
  3. Sep 23, 2009 #2
    I didn't find the result but I think you don't have the right thing to prove (the last x is an x^3).


    (But I may be wrong).
     
  4. Sep 23, 2009 #3

    Dick

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    Write I_n as integral of x*(1-x^3)^n=x*(1-x^3)*(1-x^3)^(n-1)=x(1-x^3)^(n-1)-x^4(1-x^3)^(n-1). You can express the pesky x^4(1-x^3)^(n-1) integral in terms of I_n and I_(n-1).
     
  5. Sep 25, 2009 #4
    Ok, I get it now. Thanks(:
     
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