Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$
where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec x)=\epsilon_{ijk} \Big[\int \,\,d^3x \,\, \vec x\times \vec a(\vec x)\Big]_k-ib \int\,\,d^3x\,\, x_ix_jc(\vec x)$$?
Context/Interpretation: $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ is obtained from $$\nabla \cdot [\vec a(\vec x) \exp(-ibt)]= {\partial\over\partial t}[c(\vec x)\exp(-ibt)]$$ which can be interpreted as a conservation law. I don't have any more context information...
Consider the form $\vec \nabla \cdot (\vec x \times \vec a)$ as it is a special case of the scalar triple product.
Or just integrate and use Stokes' theorem on the form $\int \vec \nabla \cdot \vec a$. I believe, depending on the choice of $\vec a$ that this could be interpreted as one of Maxwell's equations.
You also may want to consider clarifying the notation $\Big[\int \,\,d^3x \,\, \vec x\times \vec a(\vec x)\Big]_k$ although I am pretty sure the Levi-Civita symbol arises due to a change in coordinate basis here.
Depending on the meaning of the notation, Gauss' law may be applicable or even http://en.wikipedia.org/wiki/Divergence#Decomposition_theorem