<s>[INST] Does $\int _0^1 \cfrac{x}{x+\cfrac{x^{2}}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$ exist? $$\int_0^1 \cfrac x {x+\cfrac{x^2}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$$
Does this integral exist?
Any hint will be appreciated.
 [/INST] $\int _{0}^{1} \frac{x}{x＋\frac{x^{2}}{x^2＋\frac{x^{3}}{x^{3}＋…}}}dx$
Let's call that monstrous integrand $u$. If you notice, $u=\frac{x}{x+xu}=\frac{1}{1+u}$. 
$u+u^2=1$, so $u^2+u-1=0$. This means $u=\frac{\sqrt{5}-1}{2}$. 
$\int_0^1 \frac{\sqrt{5}-1}{2} \,dx = \boxed{\frac{\sqrt{5}-1}{2}}$
</s>