Thread: How 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

Originally Posted by noppawit

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

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.

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.