please help im unable to evaluate this integral

[tepid gazelle]

It is supposed to use the Beta function

[stiff lance]

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[tepid gazelle]

$\beta (m, n)= \int_0^{\infty} \frac{x^{n-1}}{(1+x)^{m+n}} dx$

this form

[teal quail]

diviirockgod6

[tepid gazelle]

i identified the integral as Beta(-3/2, 11/2)

but i am not getting any sqrt(2) in my answer..

[tiny rivet]

Well, you have x^(9/4)/(1 + x)^4. So, n - 1 = 9/4 and m + n = 4.

Ah, you want to evaluate it, too.

[tepid gazelle]

oh wait

9/4

i wrote 9/2 💀

mb

wait lemme try again

beta(3/4, 13/4)

yeah please help i dont know

how to evaluate this

[tiny rivet]

Well, let's see.
Β(3/4, 13/4) = Γ(3/4)Γ(13/4)/Γ(4)
Γ(4) is 6.
For the numerator we do the following:
Γ(3/4)Γ(13/4) = (9/4)Γ(3/4)Γ(9/4) = (9/4)(5/4)Γ(3/4)Γ(5/4) = (9/4)(5/4)(1/4)Γ(3/4)Γ(1/4)
Then we recall that Γ(x)Γ(1 - x) = π/sin(πx) for 0 < x < 1, so:
Γ(3/4)Γ(13/4) = (9/4)(5/4)(1/4)π/sin(π/4) = 45π√(2)/64
So:
Β(3/4, 13/4) = (45π√(2)/64)/6 = 15π√(2)/128

[tepid gazelle]

OHH, the euler's reflection formula
Thanks a lot i got it now !! 👍

just started learning about beta function, gamma function an hour ago for fun

nice nice

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