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How to evaluate the integral $\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots\sqrt{1+x}}}}}dx=?$
We have :
$$\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+x}}}dx=\frac{8}{315}\sqrt{2}\Big(16+\sqrt{233+317\sqrt{2}}\Big)$$
We are lucky because this integral have an anti-derivative like here.
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An incorrect answer for an integral
As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed).
$$
''\int_{0}^{\infty}\frac{1}{\left(1+...
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