Integration Maths Problem

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  1. Daaaaaaaaaaaaaaaan's Avatar
    1 21-03-2012  20:41
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    Integration Maths Problem
    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.
    Last edited by Daaaaaaaaaaaaaaaan; 21-03-2012 at 20:43.
  2. the bear's Avatar
    2 21-03-2012  23:04
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    Re: Integration Maths Problem
    i managed to transform the last version into

    int 1/(w8 - 1) dw

    but am not sure what to do after that
    Post rating:
    1
  3. Daaaaaaaaaaaaaaaan's Avatar
    3 22-03-2012  10:01
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    Re: Integration Maths Problem
    Might I ask how you got that?
  4. latentcorpse's Avatar
    4 22-03-2012  11:20
    • Peer Of The TSR Realm
    Re: Integration Maths Problem
    Wolfram Alpha cannot do this (at least in the amount of time I am entitled to on their servers!)
  5. the bear's Avatar
    5 23-03-2012  11:00
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    Re: Integration Maths Problem
    (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
  6. rhysowen's Avatar
    6 23-03-2012  11:03
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    • Posts: 96
    Re: Integration Maths Problem
    (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
  7. the bear's Avatar
    7 23-03-2012  11:14
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    Re: Integration Maths Problem
    (Original post by rhysowen)
    I thought this forum had LaTeX support
    LaTex support sounds like something Granny would purchase from Boots
  8. rhysowen's Avatar
    8 23-03-2012  11:15
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    Re: Integration Maths Problem
  9. DFranklin's Avatar
    9 23-03-2012  11:25
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    Re: Integration Maths Problem
    (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.
    Post rating:
    2
  10. ztibor's Avatar
    10 23-03-2012  21:30
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    Re: Integration Maths Problem
    (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
    Last edited by ztibor; 23-03-2012 at 21:36.
    Post rating:
    3
  11. the bear's Avatar
    11 24-03-2012  03:41
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    Re: Integration Maths Problem
    Well done ztibor & DFranklin
  12. Daaaaaaaaaaaaaaaan's Avatar
    12 24-03-2012  14:18
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    Re: Integration Maths Problem
    Thank you
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