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Thread: Proof of a problem involving definate integrals?

  1. #1
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    Proof of a problem involving definate integrals?

    How am I supposed to approach this problem?
    I tried using integration by parts, but there was no way to eliminate the exponent...
    Maybe I need to use the odd/even function property with integrals? I am not sure how to proceed...problem.jpg

  2. #2
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    Quote Originally Posted by twohaha View Post
    How am I supposed to approach this problem?
    I tried using integration by parts, but there was no way to eliminate the exponent...
    I doubt that any one can read what you posted.
    Why not type it out?
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  3. #3
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    Ok...

    If a and b are positive numbers, show that
    (&int(from 0 to 1)xa(1-x)bdx) = (&int (from 0 to 1) xb(1-x)adx)

  4. #4
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    Quote Originally Posted by twohaha View Post
    If a and b are positive numbers, show that
    (&int(from 0 to 1)xa(1-x)bdx) = (&int (from 0 to 1) xb(1-x)adx)
    Let [tex]t=1-x[/tex] then [tex]dx=-dt[/tex] and [tex]\begin{array}{*{20}{c}} x&\| & 0&1 \\ \hline t&\| & 1&0 \end{array}[/tex].
    So [tex]\int_0^1 {{x^a}{{\left( {1 - x} \right)}^b}dx} = \int_1^0 { - {{\left( {1 - t} \right)}^a}{t^b}dt} [/tex]
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  5. #5
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    Quote Originally Posted by pka View Post
    Let [tex]t=1-x[/tex] then [tex]dx=-dt[/tex] and [tex]\begin{array}{*{20}{c}} x&\| & 0&1 \\ \hline t&\| & 1&0 \end{array}[/tex].
    So [tex]\int_0^1 {{x^a}{{\left( {1 - x} \right)}^b}dx} = \int_1^0 { - {{\left( {1 - t} \right)}^a}{t^b}dt} [/tex]
    What does [tex]\begin{array}{*{20}{c}} x&\| & 0&1 \\ \hline t&\| & 1&0 \end{array}[/tex].mean?

    Is this a type of integral? My knowledge in calculus isn't very advanced...

  6. #6
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    Quote Originally Posted by twohaha View Post
    What does [tex]\begin{array}{*{20}{c}} x&\| & 0&1 \\ \hline t&\| & 1&0 \end{array}[/tex].mean?
    Do you understand change of variables?
    As we change from x to t we have to change the limits of integration.
    That is what that chart is about.
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

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