Find all continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $(f(x))^2+8=\int\limits_0^xf(t)dt$

Question

Find all continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$(f(x))^2+8=\int\limits_0^xf(t)dt$$

If I set $F(x)=\int\limits_0^xf(t)dt$ then $F$ is differentiable and $F'(x)=f(x)$,

so $(f(x))^2+8$ is diferentible as it does $\sqrt{F(x)-8}=|f(x)|$,

and then $F'(x)=(|f(x)|^2+8)'=2|f(x)||f(x)|'=f(x)$.

and then I don't know how to continue.

Answer 1

$$(f(0))^2+8=\int\limits_0^0f(t)dt=0\Rightarrow(f(0))^2=-8$$ which is a contradiction since $f$ is real valued.