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Please help me with this, I tried induction but didn't succeed. I really have no idea what else to try.

Prove or refute the equation(m and n can be any natural number):

\begin{array}{cc} \int_{0}^{1} \ x^m (1-x)^n\,dx = \int_{0}^{1}\ x^n (1-x)^m\,dx \\ \end{array}

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    It's the same trick as a standard text book example/exercise of proving$$\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$$ It can be verified by letting $u=a-x$. – 2017-02-07
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    Duplicate of http://math.stackexchange.com/questions/2132500/prove-that-int-01xa-left1-x-rightbdx-int-01xb-left1-x-right – 2017-02-07

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Just consider the substitution $s=1-x$ to see that the equation holds.

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    It really worked. It was much easier than I thought. Thank you. – 2017-02-07
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Use integration by substitution: let $1-x=t$. \begin{align*} \int_0^1 x^m(1-x)^ndx&=-\int_1^0 (1-t)^mt^ndt\\ &=\int_0^1 (1-t)^mt^ndt\\ &=\int_0^1 (1-x)^mx^ndx \end{align*}