# How to prove $\int\limits_0^1 {{x^m} \times {{(1 - x)}^n}dx} = \int\limits_0^1 {{x^n} \times {{(1 - x)}^m}dx}$

I am studying Apostol's Calculus Vol1. And in page 217 , I have trouble with the question 24. The problem is:
If m and n are positive integers, show that:
$\int\limits_0^1 {{x^m} \times {{(1 - x)}^n}dx} = \int\limits_0^1 {{x^n} \times {{(1 - x)}^m}dx}$

## my work

I make the question into 3 different situations: m = n, m > n, m < n (the third situation is the same as the second).I cannot prove the second situation.Here is my try: