I need to find continuous positive Functions $f$ and $g$ and we have $F(x)=\int\_0^xf(t)dt$ and $G(x)=\int\_0^x g(t)dt$ such that $f(x)=o(g(x))$ does not imply $F(x)=o(G(x))$.

Here $f(x)=o(g(x))$ means $\lim\_{x\rightarrow\infty}f(x)/g(x)=0$

I feel this would work with some logarithmic functions but I could not find out the exact functions to satisfy it.