Given that m and n are positive integers, show that

int [from 0 to 1] (x^m)(1-x)^n dx = int [from 0 to 1] (x^n)(1-x)^m dx

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- Aug 16th 2008, 05:59 PMnoppawitHow to show that it's equal (Intergral)
Given that m and n are positive integers, show that

int [from 0 to 1] (x^m)(1-x)^n dx = int [from 0 to 1] (x^n)(1-x)^m dx - Aug 16th 2008, 07:54 PMmr fantastic
There are numerous approaches to show that each integral is equal to . Here is one approach:

Using integration by parts:

.

In fact, using integration by parts t times you get:

.

Let t = n:

.

The final integration is simple.

From the symmetry of the answer, it's clear that has the same value. - Aug 17th 2008, 12:27 AMCoomast
If you use the substitution:

on the left hand side, you will almost immediately find the right hand side of the equality. This might be a bit easier than mr fantastic's (perfectly valid) solution.