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    Let's see who's got the brain to solve this problem.
    Compute the integral
    

\displaystyle I= \int \sqrt{2+\sqrt{2+...+\sqrt{2+x}}}  dx
    where the expression in the integrand contains n\geq 1 square roots.
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    (Original post by Omio)
    Let's see who's got the brain to solve this problem.
    THe integral may be written form bellow
    \displaystyle \int \sqrt{\sqrt{\sqrt{....\sqrt{x+2}  +2}+2}+2}\ dx
    And can be solved with series of substitution
    First:
    a=x+2 and dx=da
    so
    \displaystyle \int \sqrt{\sqrt{....\sqrt{\sqrt{a}+2  }+2}+2}\ da
    now substitute
    b=Va ->a=b^2 ->da=2b db so
    \displaystyle 2\int b\sqrt{\sqrt{....\sqrt{b+2}+2}+2  }\ db
    substitute
    c=\sqrt{b+2} -> b=c^2-2 ->db=2c dc
    so
    \displaystyle 4\int c\cdot (c^2-2)\sqrt{\sqrt{....\sqrt{c+2}+2}+  2}\ dc
    and continue with similar substitution up to there is not sqare root.
    You will for example substituting n-times (n square root) and at last with z
    \displaystyle 2^n\int z(z^2-2)((z^2-2)^2-2)\cdot ..... \cdot (((..(z^2-2)^2....)^2-2)^2-2)\ dz
    Then multplying the powers integrate by terms and resubstitute.
    It means long and systematic work.
    For an example:
    http://www.wolframalpha.com/input/?i...29%29%29%29+dx
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    (Original post by Omio)
    Let's see who's got the brain to solve this problem.
    It would be better, If you post this in here
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    see if you can get an ODE out of that by differentiating and squaring
 
 
 
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