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If $ a $ and $ b $ are positive numbers, show that

$$ \int^1_0 x^a(1 - x)^b \,dx = \int^1_0 x^b(1 - x)^a \,dx $$

$$\int_{0}^{1} x^{b}(1-x)^{a} d x$$

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Justin I.

January 31, 2021

If f is continuous on [0, ????], use the substitution u = ???? ? x to show that

Missouri State University

Oregon State University

Harvey Mudd College

Boston College

So we're going to show that this following equation is true. When A and B are positive numbers in order to do that first we're going to let a equal one and be equal to two. We're going to solve for the first part of our equation. So we're going to do are integral From 0 to 1 of x rays, that one times one minus X. All this, raised it to the X. So next year you're going to want to factor this and break this down our middle portion so that we end up with x minus X squared sorry not X squared two X squared plus X cubed dx. So when we take the integral of each one of these individual parts, we're going to end up with one half minus two cubed plus 1/4 Which equals 1/12. All right. So now that we've shown that the second half or sorry the first half of the equation is equal to one half. We're going to do the same thing for the second half of the equation. Mhm. So we have the integral of 0-1 X squared Parentheses, -X. Um dx So then we're going to rewrite this integral of 01 as X squared minus execute and we get this from distributing the X square to our parentheses the X. Now when we take the integral of each one of our components Here we end up with 1/3 minus one force. This is after you take the integral of each component and plug in your one. This will also give us 1/12 showing that our equation is indeed true when our A and B is possible.

Georgia Southern University