# proof of integral

• May 2nd 2007, 10:51 PM
Recklessid
proof of integral
Hi, can someone help me with this:

If m and n are positive integers, show that

int_{0}^{1} (x^m)(1-x)^n dx (Thats the integral from 0 to 1)

is equal to

int_{0}^{1} (x^n)(1-x)^m dx.

Thanks a bunch.
• May 2nd 2007, 11:16 PM
ecMathGeek
Quote:

Originally Posted by Recklessid
Hi, can someone help me with this:

If m and n are positive integers, show that

int_{0}^{1} (x^m)(1-x)^n dx (Thats the integral from 0 to 1)

is equal to

int_{0}^{1} (x^n)(1-x)^m dx.

Thanks a bunch.

INT{0,1} x^m*(1 - x)^n dx = INT{0,1} x^n*(1 - x)^m dx

LHS: INT{0,1} x^m*(1 - x)^n dx
Let u = 1 - x <--> du = -1 dx --> dx = -1 du

The limits of integration become:
x = 0 --> u = 1 - 0 = 1
x = 1 --> u = 1 - 1 = 0

- INT{1,0} (1 - u)^m*u^n du

Switching the limits of integration back to {0,1} reverses the sign of the integration:

INT{0,1} u^n*(1 - u)^m du

Setting the LHS and RHS equal, we see that:

INT{0,1} u^n*(1 - u)^m du = INT{0,1} x^n*(1 - x)^m dx

Since both sides are the integration of the same function (though in terms of seperate variables) with the same limits of integration, the integrations are the same. The variable used in integration does not matter so long as the function and the limits are the same.
• May 3rd 2007, 06:23 AM
ThePerfectHacker
They are the same because they are equal to Beta function.
• May 3rd 2007, 07:33 AM
ecMathGeek
Quote:

Originally Posted by ThePerfectHacker
They are the same because they are equal to Beta function.

I think he's in either Calculus 1 or Calculus 2. In either case, I doubt he would have seen the Beta function before. Interesting observation though.
• May 3rd 2007, 09:59 AM
ThePerfectHacker
Quote:

Originally Posted by ecMathGeek
I think he's in either Calculus 1 or Calculus 2. In either case, I doubt he would have seen the Beta function before. Interesting observation though.

Some people are familar with Beta and Gamma functions when they are in Calculus II.
• May 3rd 2007, 10:23 AM
ecMathGeek
Quote:

Originally Posted by ThePerfectHacker
Some people are familar with Beta and Gamma functions when they are in Calculus II.

No doubt that's possible. That's why I said "I doubt he would have seen the Beta function before." But if his teacher felt like showing his class some interesting things that can be done with integrations, then maybe he would have seen it before.