A converse theorem to $\operatorname{div} (\operatorname{curl}(\mathbf{F}))=0$

Question

I have already proved the following: Let $\mathbf{F}$ be a vector field of class $C^2$ defined on $ \mathbb{R}^3$. If $\nabla\cdot\mathbf{F}=0$, then there is some vector field $\mathbf{G}$ such that $\mathbf{F}=\nabla\times\mathbf{G}$. The book suggests to use:$$\mathbf{G}(x,y,z)=\int_0^1t\mathbf{F}(xt,yt,zt)\times\mathbf{r}\,dt$$ where $\mathbf{r}=(x,y,z).$ One question arises: how on earth has it found such a vector field?