Prove an identity of integral equations

by RoRi

October 19, 2015

For positive integers $m, n$ prove

$\int_0^1 x^m (1-x)^n \, dx = \int_0^1 x^n (1-x)^m \, dx.$

Proof. Let

$u = 1-x \qquad \implies \qquad du = -dx.$

Then,

$\begin{align*} \int_0^1 x^m (1-x)^n \, dx &= - \int_{u(0)}^{u(1)} (1-u)^m u^n \, du \\ &= -\int_1^0 (1-u)^m u^n \, du \\ &= \int_0^1 (1-u)^m u^n \, du \\ &= \int_0^1 x^n (1-x)^m \, dx. \qquad \blacksquare \end{align*}$

Stumbling Robot

Prove an identity of integral equations

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