Prove two integrals are the same using U substitution

[v0id19]

1. The problem statement, all variables and given/known data
If a and b are positive numbers, show that \int_0^1 x^a*(1-x)^b\,dx = \int_0^1 x^b*(1-x)^a\,dx using only U substitution.


2. Relevant equations
Just U substitution and the given equation--I can't use multiplication rules or anything like that; otherwise it would be easy.


3. The attempt at a solution
I tried to set U=(1-x) and I end up with \int_0^1 (1-U)^a*(U)^B\,dx for the right side, but that doesn't seem to get me anywhere. I know I somehow need to switch the places of the x and (1-x) but I can't seem to get around going in a circle and ending up with what I started with.

[Pere Callahan]

Multiplication of real numbers is commutative, so


(1-U)^a*U^b=U^b*(1-U)^a


Or does * denote something different from multiplication of real numbers?

If you let U = 1-x you also have dx=-dU. Morover your integration bounds change. You need to be more careful in applying the u substitution rule.

[v0id19]

If you let U = 1-x you also have dx=-dU. Morover your integration bounds change.
yeah I just realized that, thanks
\int_1^0 (1-U)^a*(U)^b\,(-du) = \int_0^1 (1-U)^a*(U)^b\,(du) but I still get stuck there. Why would switching the order of (U-1)a and Ub help?

[Pere Callahan]

Haven't you just shown


\int_0^1{x^a(1-x)^bdx}=\int_0^1{u^b(1-u)^adu}

?

How does this relate to the original question you are asked to prove? Remeber, that it doesn't make a difference how you call the integration variable. x, u, whatever, it is only a label.

[v0id19]

Oh. Wow. That makes sense!

Thanks a bunch!! :biggrin: