Let so this is and
The definition of the beta function reads
so that and hence
so
Using that
my answer is
which is not yours either
the integral from 0-1 of √(1-x^4) dx = (Γ(1/4)^2)/(6√2π)
Here is how i tried to tackle it...
let t= x^4 , then dt/dx = 4x^3. And also x^3 = t ^(3/4).
Now the integral from 0-1 of (1-t) ^1/2 dt/4x^3
1/4 of the integral from 0-1 of x^(-3) (1-t)^1/2 dt
1/4 of the integral from 0-1 of. t^(-3/4) (1-t) ^1/2 dt
x-1 = -3/4 therefore x= 1/4
y-1 = 1/2 then. y= 3/2
which is 1/4 B(1/4 , 3/2) NOT equal to the LHS of the equation.
i also tried it by substituting x^2 = sin u and end up with a complete different answer of B(1/2, 3/2).
can some one pls point out what I missed.
Any help would be much appreciated
Thanks Ruun
I think we have the same answer if it wasn't for the following
when you work out for x, you did x-1 = 3/4, but it should have been (-3/4) , which gives x = 1/4
But either way the answer should be equal to Γ(1/4)^2/(6√2π).....its 6 root 2pie in the denominator ,
Hi Abeyatta,
I agree with your answer of
It can be converted to the given answer by the use of a few identities.
Start with
That gets you a in the denominator and you deal with that by using the so called duplication formula
(Substitute x = 5/4).
Remove the and by use of the identities and finally
(The also requires the use of