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Math Help - integral problem

  1. #1
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    integral problem

    please help me with this integral problem:

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

    substitution method

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

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

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

    ... You derive immediately that is...

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

    ... so that is...

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

    Kind regards

    \chi \sigma
    Last edited by chisigma; July 22nd 2010 at 02:08 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 y= \sqrt {x + \sqrt{x + \sqrt{x + ...}}} (1)

    ... You derive immediately that is...

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

    ... so that is...

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

    Kind regards

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

    ... it seems to be that...

    a) is y(0)=0 ...

    b) y(x) is increasing with x...

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

    y^{2} - y - x =0 (2)

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

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

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

    \chi \sigma
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