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Math Help - substitution

  1. #1
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    Post substitution

    Hello agian.

    how do u do the substitution t = x ^ 1-2

    for dx / 3 + square root x

    with upper limit as 1
    and lower as 0

    Thank you.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by simon
    Hello agian.

    how do u do the substitution t = x ^ 1-2

    for dx / 3 + square root x

    with upper limit as 1
    and lower as 0

    Thank you.
    \int_0^1 \frac{dx}{3+\sqrt x}

    Now set:
    t = \sqrt x
    Thus
    dt = \frac{1}{2} \frac{1}{\sqrt x} dx
    or
    dx = 2 \sqrt x dt = 2 t \, dt

    For the limits on the integral:
    Upper limit: t = \sqrt 1 = 1
    Lower limit: t = \sqrt 0 = 0

    Thus:
    \int_0^1 \frac{dx}{3+\sqrt x}
    = \int_0^1 \frac{2 t \, dt}{3 + t} =2 \int_0^1 dt \frac{t}{t+3}=2 \int_0^1 dt \left ( 1 - \frac{3}{t+3} \right )
    =2t|_0^1 - 6 \, ln|t+3| |_0^1 = 2 - 6 \, ln4 + 6 \, ln 3
    = 2 + 6 \, ln \left ( \frac{3}{4} \right )

    -Dan
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by topsquark
    \int_0^1 \frac{dx}{3+\sqrt x}

    Now set:
    t = \sqrt x
    Thus
    dt = \frac{1}{2} \frac{1}{\sqrt x} dx
    or
    dx = 2 \sqrt x dt = 2 t \, dt

    For the limits on the integral:
    Upper limit: t = \sqrt 1 = 1
    Lower limit: t = \sqrt 0 = 0

    Thus:
    \int_0^1 \frac{dx}{3+\sqrt x}
    = \int_0^1 \frac{2 t \, dt}{3 + t} =2 \int_0^1 dt \frac{t}{t+3}=2 \int_0^1 dt \left ( 1 - \frac{3}{t+3} \right )
    =2t|_0^1 - 6 \, ln|t+3| |_0^1 = 2 - 6 \, ln4 + 6 \, ln 3
    = 2 + 6 \, ln \left ( \frac{3}{4} \right )

    -Dan
    Please award yourself the Math Help Forum Medal for clairvoyance, for
    exceptional performance in interpreting strangely/obscurely phrased
    questions

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack
    Please award yourself the Math Help Forum Medal for clairvoyance, for
    exceptional performance in interpreting strangely/obscurely phrased
    questions

    RonL
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack
    Please award yourself the Math Help Forum Medal for clairvoyance, for
    exceptional performance in interpreting strangely/obscurely phrased
    questions

    RonL
    (Everybody: Don't tell CaptainBlack, but I knew he was going to say that! )

    -Dan
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  6. #6
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    CaptainBlack, CaptainBlack! Did you hear' what topsquark-the fizizist said 'bout you! Do not tell him I said that...
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by topsquark
    (Everybody: Don't tell CaptainBlack, but I knew he was going to say that! )

    -Dan
    I suppose you are now expecting to be awarded Oak Leaves to your medal.

    RonL

    (Fishing for an award of the Math Help Form Medal for obscure humour )

    RonL
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack
    I suppose you are now expecting to be awarded Oak Leaves to your medal.

    RonL

    (Fishing for an award of the Math Help Form Medal for obscure humour )

    RonL
    (Snort!) I'll be more likely to get the Poison Oak Leaves Award...

    -Dan
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