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Thread: integral problem

  1. #1
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    integral problem

    please help me with this integral problem:

    $\displaystyle \int(x+(x+(x+....)^\frac{1}{2})^\frac{1}{2})^\frac {1}{2}dx=....$

    substitution method

    $\displaystyle (x+(x+(x+....)^\frac{1}{2})^\frac{1}{2})^\frac{1}{ 2}=y$

    $\displaystyle x+(x+(x+....)^\frac{1}{2})^\frac{1}{2}=y^2$

    $\displaystyle x+y=y^2$

    ??????
    i've no idea what the next step are.....
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting...

    $\displaystyle \displaystyle y= \sqrt {x + \sqrt{x + \sqrt{x + ...}}} $ (1)

    ... You derive immediately that is...

    $\displaystyle \displaystyle \sqrt{x + y} = y \rightarrow y = \frac{1 + \sqrt{1+4x}}{2}$ (2)

    ... so that is...

    $\displaystyle \displaystyle \int \sqrt {x + \sqrt{x + \sqrt{x + ...}}}\ dx = \int \frac{1 + \sqrt{1+4x}}{2}\ dx$ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Jul 22nd 2010 at 01:08 AM. Reason: not sure of the sigh in (2) and (3)... see my succesive post...
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  3. #3
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    Quote Originally Posted by chisigma View Post
    Setting...

    $\displaystyle \displaystyle y= \sqrt {x + \sqrt{x + \sqrt{x + ...}}} $ (1)

    ... You derive immediately that is...

    $\displaystyle \displaystyle \sqrt{x + y} = y \rightarrow y = \frac{1 - \sqrt{1+4x}}{2}$ (2)

    ... so that is...

    $\displaystyle \displaystyle \int \sqrt {x + \sqrt{x + \sqrt{x + ...}}}\ dx = \int \frac{1 - \sqrt{1+4x}}{2}\ dx$ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    i see...

    thanks chisigma.... i'll continue from here....
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  4. #4
    MHF Contributor chisigma's Avatar
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    The question has a little minor problem: setting...

    $\displaystyle \displaystyle y(x) = \sqrt {x + \sqrt{x + \sqrt{x + ...}}}$ (1)

    ... it seems to be that...

    a) is $\displaystyle y(0)=0$ ...

    b) $\displaystyle y(x)$ is increasing with x...

    Now is a little difficult to extablish which of the two solutions of the equation...

    $\displaystyle y^{2} - y - x =0$ (2)

    ... meets both the requirements a) and b), because...

    c) $\displaystyle \displaystyle y(x)= \frac{1 - \sqrt{1+4x}}{2}$ vanishes in $\displaystyle x=0$ but is decreasing with x...

    d) $\displaystyle \displaystyle y(x)= \frac{1 + \sqrt{1+4x}}{2}$ is increasing with x but is $\displaystyle y(0)=1$ ...

    Since it is requested fo find a primitive of (1) and all primitives are defined unless a constant, in my opinion the solution is...

    $\displaystyle \displaystyle \int \sqrt {x + \sqrt{x + \sqrt{x + ...}}}\ dx = \int \frac{1 + \sqrt{1+4x}}{2}\ dx $ (3)

    ... even if I'm not 'cent per cent' sure ...

    $\displaystyle \chi$ $\displaystyle \sigma$
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