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Math Help - Another "theoretical" thing about integrals I have no idea how to do

  1. #1
    s3a
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    Another "theoretical" thing about integrals I have no idea how to do

    Question:
    If a and b are positive numbers, show that

    this(integral (x^a * (1-x)^b) from 0 to 1) = this(http://www.wolframalpha.com/input/?i=integral+(x^b+*+(1-x)^a)+from+0+to+1).

    And for this problem, I don't even have the answer in the back of my book! Someone please help!


    (Ignore what Wolfram Alpha says - I am just using those links to make it more visual because I don't know how to use latex)

    Any help would be GREATLY appreciated!
    Thanks in advance!
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  2. #2
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    Let u = 1-x and du = -dx and you should get:
     \int_0^1 x^a (1-x)^b dx = -\int_1^0 (1-u)^a u^b du = \int_0^1 (1-u)^a u^b du
    And since we're dealing with a definite integral, it doesn't matter that we ended up with u's.
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  3. #3
    s3a
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    Can you please explain why we assume equality of the variables u and x in the last step? Just stating that we can do so because we're dealing with a definite integral doesn't explain it well enough for me.
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  4. #4
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    Since it's a definite integral, you ultimately wind up plugging the limits into the antiderivative. So it doesn't matter if you start with x or u, as whichever variable you use will be replaced by the limits 0 and 1 anyway.

    If you aren't buying it, just take your favorite definite integral and change the variable. You should get the same answer.
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