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:
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In fact, using integration by parts t times you get:
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Let t = n:
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The final integration is simple.
From the symmetry of the answer, it's clear that has the same value.