$$S = \frac{1}{\int_1^2 \frac{1}{\int_1^2 \frac{1}{\int_1^2 \ddots}}}$$
I know that these kind of recursive questions are one inside another - meaning that:
$$ S = \frac{1}{\int_1^2 S}$$
The question is what is the variable? $x$ ? $ S$ ? whatever I would like it to be?
If the variable is $x$ (or anything else than $S$ ) then the answer is:
$$ S (x \mid _1^2) = S = 1 \Rightarrow S = 1$$
If the variable is $S$ than the answer is:
$$ S ( \frac{S^2}{2} \mid _1^2 ) = S[ 2 - 0.5] = 1.5S = 1 \Rightarrow S = \frac{2}{3}$$
Which is correct? I assume there is only one value that is correct and not $2$ because, this integral keeps on forever so we don't actually know if it is $dx$ or $dS$
$$S = \frac{1}{\int_1^2 \frac{1}{\int_1^2 \frac{1}{\int_1^2 \ddots}}}$$ if the right side is anything, it is a value at least in $[-\infty,+\infty]$, so $S$ is not a variable. Said this I would procede this way:
$$S = \frac{1}{\int_1^2 Sdx} \iff S^2 = \frac{1}{\int_1^2 dx}= 1 \iff S=\pm 1.$$ I actually have no idea if the negative solution is acceptable.