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Technical question on integrals.

  1. Sep 3, 2006 #1

    MathematicalPhysicist

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    i have to show that:
    1)[tex](-1)^n\int_{-1}^{1}(x^2-1)^ndx=2^{2n+1}(n!)^2/(2n+1)![/tex]
    2) [tex]\binom{n}{k}=[(n+1)\int_{0}^{1}x^k(1-x)^{n-k}dx]^{-1}[/tex]

    for the first part i thought to use newton's binomial, i.e:
    [tex](1-x^2)^n=\sum_{k=0}^{n}\binom{n}{k}(-x^2)^k[/tex]
    but it didn't get me far.
    for the second part i dont have a clue, i dont think you can integrate the integral by parts or substitution can you?
     
  2. jcsd
  3. Sep 3, 2006 #2

    StatusX

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    Where'd you get stuck after using the binomial theorem? Just integrate term by term.
     
  4. Sep 3, 2006 #3

    MathematicalPhysicist

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    this is what i got:
    2-2n/3+2n(n-1)/10+...+2(-1)^n/(2n+1)
    i don't know how to procceed from here.
     
  5. Sep 3, 2006 #4
    yeah, i think number 2 can be worked out using integral by parts... the tabular method.
    all except the last term of the resulting series is zero when the bounds are substituted in.
     
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