Beta function and changing variables

  1. 1. The problem statement, all variables and given/known data
    [tex]\int_{0}^{3}\sqrt[3]{\frac{3-x}{x^2}}\: \mathrm{d}x[/tex]

    2. The attempt at a solution
    I'm pretty sure I have to use Euler's Beta function, so I tried to change the limits to 0 and 1 by setting x = 3·u (so dx = 3·du). However there must be some mistake when I did it because I checked and I did it wrong:
    [tex]\int_{0}^{3}\sqrt[3]{\frac{3-x}{x^2}}\: \mathrm{d}x=\int_{0}^{1}\sqrt[3]{\frac{3-3u}{(3u)^2}}\: 3\cdot\mathrm{d}u=3^{\frac{2}{3}}\cdot\beta\left(\frac{1}{3},\frac{4}{3}\right)[/tex]

    Thanks a lot for your help.
     
    Last edited: Feb 14, 2009
  2. jcsd
  3. CompuChip

    CompuChip 4,297
    Science Advisor
    Homework Helper

    Hmm then what should be the correct answer. Because, unless I am still asleep, both your change of variables and your identifying the arguments of the beta function are quite all right. I even got the same prefactor with a 2/3 exponent.
     
  4. Well, Mathematica says:
    [tex]\int_{0}^{3}\sqrt[3]{\frac{3-x}{x^2}}\: \mathrm{d}x=\left(\frac{3}{2}\right)^{\frac{2}{3}}\cdot\beta\left(\frac{1}{3},\frac{1}{2}\right)[/tex]
     
  5. CompuChip

    CompuChip 4,297
    Science Advisor
    Homework Helper

    But when I evaluate both numerically, I get the same answer up to 4*$MachineEpsilon.

    Code (Text):
    Beta[1/3, 4/3] == (1/2)^(2/3) Beta[1/3, 1/2] // FullSimplify
    gives True.

    So probably there is some property of the Beta function (which can most likely be derived from the Gamma function) which you need to show.
     
  6. OK thanks I think Mathematica isn't using beta function though because in all my integrals when I try to check them I get a different output, which I can't get to even when I try FullSimplify[Beta[whatever, whatever]]. However after try the comparison I get true.
     
  7. CompuChip

    CompuChip 4,297
    Science Advisor
    Homework Helper

    Just a general remark...

    In my experience Mathematica sometimes does strange things with comparisons, and seen instances of
    SomeExpression == DifferentExpression
    giving True.

    I suggest always FullSimplify'ing the difference betweeen your and Mathematica's answer
    MyExpression - MathematicaResult // FullSimplify
    and checking it gives zero.
     
  8. Thanks, I'm just learning how to use Mathematica and I didn't know about that feature.

    Now I'm stuck with the last integral in my problem sheet... this one I can't solve (b > 2 and a > 0).
    [tex]\int_{0}^{\infty}\frac{\sqrt[3]{x}-\sqrt{x}}{x^b-a^b}\: \mathrm{d}x[/tex]
    I'm thinking that it's going to be a beta too (duh, it's the gamma-beta functions section of the problem set), so I should change that infinity to either 1 or pi/2.
    Since changing it to pi/2 would imply having arctan in my integral (wouldn't it?) it doesn't look good. So I guess I need a 1... but I can't think of anything good there.
     
    Last edited: Feb 15, 2009
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