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Integration Maths Problem

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    Can anyone help me solve:
    Int (sqrt(x+1)-sqrt(x))^(3/2) dx.
    With substitution, it becomes of the form:

    Int (cosh(y)-sinh(y))^(3/2)/sinh(2y) dy
    or 1//2 Int exp(-3y/2)/(exp(2y)-exp(-2y)) dy

    Even if you cannot integrate this function, ideas would be extremely helpful.
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    i managed to transform the last version into

    int 1/(w8 - 1) dw

    but am not sure what to do after that
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    Might I ask how you got that?
  4. Offline

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    Wolfram Alpha cannot do this (at least in the amount of time I am entitled to on their servers!)
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    (Original post by Daaaaaaaaaaaaaaaan)
    Might I ask how you got that?
    i cannot reveal all the details due to TSR policy... but basically you let

    w = ey/2
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    (Original post by Daaaaaaaaaaaaaaaan)
    Can anyone help me solve:
    Int (sqrt(x+1)-sqrt(x))^(3/2) dx.
    With substitution, it becomes of the form:

    Int (cosh(y)-sinh(y))^(3/2)/sinh(2y) dy
    or 1//2 Int exp(-3y/2)/(exp(2y)-exp(-2y)) dy

    Even if you cannot integrate this function, ideas would be extremely helpful.
    I thought this forum had LaTeX support
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    (Original post by rhysowen)
    I thought this forum had LaTeX support
    LaTex support sounds like something Granny would purchase from Boots
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    http://www.thestudentroom.co.uk/wiki/Latex
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    (Original post by the bear)
    i managed to transform the last version into

    int 1/(w8 - 1) dw

    but am not sure what to do after that
    Assuming this is correct, you can write

    w^8 - 1 = (w^4+1)(w^4-1) = (w^4+1)(w^2+1)(w^2-1).

    Then w^4+1 = (w^2-w\sqrt{2} +1)(w^2+w\sqrt{2}+1).

    Then some (painful) partial fractioning will finish things off.
  10. Offline

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    (Original post by Daaaaaaaaaaaaaaaan)
    Can anyone help me solve:
    Int (sqrt(x+1)-sqrt(x))^(3/2) dx.
    With substitution, it becomes of the form:

    Int (cosh(y)-sinh(y))^(3/2)/sinh(2y) dy
    or 1//2 Int exp(-3y/2)/(exp(2y)-exp(-2y)) dy

    Even if you cannot integrate this function, ideas would be extremely helpful.
    For  \displaystyle \int (\sqrt{x+1}- \sqrt{x})^{\frac{3}{2}}dx
    Use the following substitution

    x=sinh^2 t
    x+1=1+sinh^2 t=cosh^2 t
    dx=2sinht \cdot cosht =sinh 2t
    So
    \displaystyle \int (cosh t - sinh t)^{\frac{3}{2}} 2sinh2t dt

    using
    \displaystyle cosht =\frac{e^t+e^{-t}}{2}
    \displaystyle sinht =\frac{e^t-e^{-t}}{2}

    the
    \displaystyle (cosh t-sinht)^{\frac{3}{2}}=\left (\frac{1}{2} \left (e^t+e^{-t}-e^t+e^{-t}\right )\right )^{\frac{3}{2}}=e^{-\frac{3}{2}t}

    So
    \displaystyle \frac{1}{2} \int e^{-\frac{3}{2}t} \cdot  \left (e^{2t}-e^{-2t}\right ) dt =
    \displaystyle \frac{1}{2} \int \left (e^\frac{t}{2} - e^{-\frac{7}{2}t}\right ) dt

    Now You can integrate this
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    Well done ztibor & DFranklin
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    Thank you

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