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Find this ratio

  1. Jul 11, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    If [itex]I_n = \displaystyle \int^1_0 x^n. \sqrt{1-x^2} dx [/itex] then
    [itex] \lim_{n \to \infty} \dfrac{I_n}{I_{n-2}}[/itex] is equal to

    2. Relevant equations

    3. The attempt at a solution
    Integrating by parts

    [itex]x^n \displaystyle \int \sqrt{1-x^2}dx - \int nx^{n-1} \int \sqrt{1-x^2} dx [/itex]

    But integrating further is useless.
     
    Last edited: Jul 11, 2013
  2. jcsd
  3. Jul 11, 2013 #2
    You are integrating it the wrong way. Keep in mind that [itex]\displaystyle \lim_{n\to\infty} \frac{I_n}{I_{n-2}} = \lim_{n\to\infty} \frac{I_{n+2}}{I_{n}}[/itex].
     
  4. Jul 12, 2013 #3
    Ignore the above post, I figured out it does not lead to the solution; so you have to resort to more analytic methods. Substituting [itex]u=x^2[/itex] in this integral gives [itex]\displaystyle \frac{1}{2}\int^{1}_{0} u^{(n-1)/2} (1-u)^{1/2}\,\,du[/itex], which is in the form of the general integral [itex]\displaystyle B(a,b) = \int^{1}_{0}x^{a-1}(1-x)^{b-1}\,dx[/itex]. This function is called the Beta function and it happens to satisfy [itex]\displaystyle B(x,y) = \frac{(x-1)!(y-1)!}{(x+y-1)!}[/itex] for integer x and y.

    Can you solve it from here?

    Note: You might get fractions inside factorials while solving the question. You don't have to know their values, just use the equation [itex](x+1)! = (x+1)x![/itex] to simplify the limit. The reason you will see those is that the equation for the Beta integral is true for a meromorphic Gamma function which satisfies [itex]\Gamma(x+1) = x![/itex], which extends the factorial function to all complex numbers.
     
    Last edited: Jul 12, 2013
  5. Jul 12, 2013 #4

    haruspex

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    Try leaving one of the x with the ##\sqrt{1-x^2}## term, i.e. factor it as ##x^{n-1}.x\sqrt{1-x^2}dx##, then integrate by parts very much as you did.
     
  6. Jul 12, 2013 #5

    utkarshakash

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    Your method was excellent. I just want to know how do you solve these complicated problems in a jiffy?
     
  7. Jul 12, 2013 #6
    Using integration by parts:
    [tex]I_n=-\frac{x^{n-1}}{3}(1-x^2)^{3/2}+\int \frac{2}{3}(n-1)x^{n-2}(1-x^2)^{3/2}[/tex]

    How do you proceed from here?

    Thanks!
     
  8. Jul 12, 2013 #7
    [tex]x^{n-2}(1-x^2)^{3/2} = (x^{n-2} - x^{n})\sqrt{1-x^2}[/tex]
    Also, don't leave out the limits.
     
  9. Jul 12, 2013 #8
    Thanks!

    @utkarshakash: Can you post the answer?
     
  10. Jul 12, 2013 #9
    Honestly, having wandered around in articles and pages involving the Gamma and Beta functions, my Beta function solution was the first one that crossed my mind. However, the one provided by haruspex is so much better if you have no prior knowledge on these topics.
     
  11. Jul 12, 2013 #10

    haruspex

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    In this case, by looking at what was to be proved. Since we wanted step the n by 2, and differentiation would only step it by one, we needed to leave one factor of x out of the differentiation.
     
  12. Jul 12, 2013 #11

    utkarshakash

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    Ah! That makes sense. Thanks.
     
  13. Jul 12, 2013 #12

    utkarshakash

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    The answer is 1.
     
  14. Jul 13, 2013 #13
    Thanks!
     
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