Beta function and changing variables

In summary, the student attempted to solve a homework problem using the Euler's Beta function, but made a mistake. After trying different limits and confirming that the answer was correct, they realized that they needed to use the inverse of the beta function.
  • #1
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Homework Statement


[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.
 
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  • #2
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.
 
  • #3
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]
 
  • #4
But when I evaluate both numerically, I get the same answer up to 4*$MachineEpsilon.

Code:
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
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1. What is the Beta function?

The Beta function, denoted by B(x, y), is a special mathematical function that is defined for positive real numbers x and y. It is used to solve various problems in statistics, calculus, and other fields of mathematics.

2. How is the Beta function related to the Gamma function?

The Beta function is related to the Gamma function through the following equation: B(x, y) = Γ(x)Γ(y) / Γ(x+y). This relationship is useful in simplifying and solving certain integrals involving the Beta function.

3. What is the use of changing variables in the Beta function?

Changing variables in the Beta function is a technique used to simplify and solve integrals involving the Beta function. It involves substituting a new variable in place of the original variable, which can often lead to a simpler expression.

4. Can the Beta function be evaluated numerically?

Yes, the Beta function can be evaluated numerically using various mathematical software and programming languages. However, for certain values of x and y, the Beta function may result in very large or small numbers, making numerical evaluation difficult.

5. What are some real-world applications of the Beta function?

The Beta function has various applications in statistics, such as in the beta distribution for modeling random variables and in Bayesian analysis. It is also used in physics, engineering, and other fields to solve problems involving probability and integrals.

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