The question is to evaluate $\iiint d^{3}r\vec{\nabla}\phi\centerdot\vec{G}$ when $\vec{\nabla}\centerdot\vec{G} = 0$
I started with $\iiint dxdydz (\frac{\partial\phi}{\partial x}G_{x}+\frac{\partial\phi}{\partial y}G_{y}+\frac{\partial\phi}{\partial z}G_{z})$
I think that $G_{x},G_{y}$ terms can be taken out of the integral as they are constants. But I don't know how to proceed beyond this point. If any of you has come across this could you please give me an idea about how to continue? Thanks.
Most of the vector differentiation identities look vaguely similar to scalar ones. For example, $$ \nabla\cdot(\phi\vec{G})=\nabla\phi\cdot \vec{G}+\phi\nabla^{\cdot}\vec{G} $$ Basically, it's the only product rule than can make sense, and it is correct. In your case, because $\nabla\cdot\vec{G}=0$, then $$ \nabla\cdot(\phi\vec{G})=\nabla\phi\cdot\vec{G}. $$ So you can trade your integral of $\nabla\phi\cdot\vec{G}$ for an integral of $\nabla\cdot(\phi\vec{G})$ instead. Because you haven't said anything about the volume over which you are integrating, I'll leave it to you to decide if you can apply the divergence theorem, and how to do that.