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How to solve this:

$\displaystyle\int \bigg(\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\sqrt{x}}}}\;\normalsize\bigg) \;dx$

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    Just let $u=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}}}}$ is OK. Then you will get $x$ in terms of polynomial of $u$. WolframAlpha canot solve it because the process time is excessed.2012-07-13

1 Answers 1

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While this is not an answer, I posted it as an answer so that I can attach the graph. On the graph you can see how close the 2 functions are. Using this fact may help.

The Black graph is the original function and the Red graph is for the approximate function.

enter image description here

A better approximation for the function $f(x)=\displaystyle\bigg(\small\sqrt{\normalsize x+\small\sqrt{\normalsize x+\sqrt{ x+\sqrt{x}}}}\;\normalsize\bigg)$ is:

$0.5(1+\sqrt[]{1+4x})$

This approximation was obtained from Nested Radicals, This formula is an exact value for the infinite case, so it may be used as an approximation only in your case.

A picture of the 2 functions is shown below. The Black graph is the original function and the Red graph is for the approximate function.

enter image description here

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    In fact I can't claim that it an optimal approximation since I came across it by plotting various curves and noticed that this one was among the better ones. The main point I wanted to show is that there may be a good approximation for the original function that may help.2011-11-24