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If $a$ and $b$ are positive numbers, show that $$\int_{0}^{1} x^{a}(1-x)^{b} d x=\int_{0}^{1} x^{b}(1-x)^{a} d x$$

$\int_{a}^{b} x^{a}(1-x)^{b} d x=\int_{a}^{b} x^{b}(1-x)^{a} d x$ Proved

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

Integrals

Section 4

The Substitution Rule

Integration Techniques

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

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this question asked us to proved the given statement given the A and B are positive numbers. So in other words, from our bounds from zero toe one, we can flip these to make our bounds from 1 to 0. Given this, we know we can, right? Negative 120 you to the B terms one minus you to the A m nd you as equivalent to from 01 you to the B times one minus you too, eh, D'You Therefore, given the fact that you could be replaced with axe, we know we can write this as from 0 to 1 X to be so we're replacing every you with acts one minus x to the power of A and R D U becomes d axe. This is proven the statement.

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