For positive integers prove

* Proof. * Let

Then,

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Stumbling Robot

A Fraction of a Dot
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Prove an identity of integral equations

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For positive integers prove

* Proof. * Let

Then,

\[ u = 1-x \qquad \implies \qquad du = -dx. \]

Then,

\begin{align*}

– \int_{u(0)}^{u(1)} (1-u)^m u^n \, du \\ &= -\int_1^0 (1-u)^n u^m \, du

\int_{u(0)}^{u(1)} (1-u)^m u^n \, du \\ &= \int_1^0 (1-u)^n u^m \, du

We can change back to x

\int_0^1 x^m (1-x)^n \, dx.\\ &= \int_0^1 x^n (1-x)^m \, dx. \qquad \blacksquare \end{align*}