Unraveling a Complicated Double Integral

[Kuma]

URGENT - Help with a double integral!

Homework Statement

This is a statistics problem for another question but it involves a somewhat complicated double integral.

So I am stuck at the last line.
We were told to use the substitution with u as i wrote down.
I don't know what to do after that part, I've tried figuring out x2 in terms of u and subbing in, but it just gets more and more complicated. The substitution with dx1 is a "correct step" as my instructor told me.

Also you need the gamma function for this:

just that the x and the y's are the alphas, and the t is x1 or x2.

Anyway so this is supposed to cancel out with the huge constant in the front resulting in a final answer of 1.

 

[lippyka]

Integral[dx1 dx2 (x1^(alpha1 - 1)) (x2^(alpha2 - 1)) exp(-x1 - x2) ] over the area specifiedHomework Equations The gamma function is Γ(x) = (x-1)!The Attempt at a SolutionLet u = x1 + x2. Then dx1 = du - dx2 and dx2 = du - dx1. We can then rewrite the integral as:Integral[du ((x1^(alpha1 - 1)) (x2^(alpha2 - 1)) exp(-u))] over (0, t) for x1 and (t-x1, ∞) for x2= Integral[du (((u-x2)^(alpha1 - 1)) (x2^(alpha2 - 1)) exp(-u))] over (0, t) for x1 and (t-x1, ∞) for x2= Integral[du ((u^(alpha1 - 1)) ((t-u)^(alpha2 - 1)) exp(-u))] over (0, t) for x1 and (t-x1, ∞) for x2= Integral[du ((u^(alpha1 - 1)) ((t-u)^(alpha2 - 1)) exp(-u))] over (0, t)= Γ(α1)Γ(α2)∫0t(u^(α1−1)(t−u)^(α2−1)exp(−u))du= Γ(α1)Γ(α2)∫0t(u^(α1+α2−1)(t−u)^(α2−1)exp(−u))du