$\begin{aligned} & F(x)=\int_0^x t \cdot f(t) d t \\ & \\ & \quad F^1(x)=x f(x) \\ & \quad \text { Given } \mathrm{F}\left(\mathrm{x}^2\right)=\mathrm{x}^4+\mathrm{x}^5, \text { let } \mathrm{x}^2=\mathrm{t} \\ & \quad \mathrm{F}(\mathrm{t})=\mathrm{t}^2+\mathrm{t}^{5 / 2} \\ & \quad F^{\prime}(t)=2 t+5 / 2 t^{3 / 2} \\ & \quad t \cdot f(t)=2 t+5 / 2 t^{3 / 2} \\ & \quad \mathrm{f}(\mathrm{t})=2+5 / 2 \mathrm{r}^{1 / 2} \\ & \quad \sum_{r=1}^{12} f\left(r^2\right)=\sum_{r=1}^{12} 2+\frac{5}{2} r \\ & \quad=24+5 / 2\left[\frac{12(13)}{2}\right] \\ & \quad=219\end{aligned}$