Given the charge density $$\rho(\vec{x})=\rho_0\delta(x_1)\delta(x_2),$$ where $\delta$ denotes the Dirac-distribution and $\vec{x}=(x_1,x_2,x_3)$, I am asked to calculate the electric field which is generated by this charge density, i.e. I have to evaluate the integral $$\int_{\mathbb{R}^3}d^3x'\frac{\rho_0\delta(x_1')\delta(x_2')}{|\vec{x}-\vec{x}'|^3}(\vec{x}-\vec{x}').$$
I am having trouble doing so! I know that
$$\int\delta(x-x')f(x)dx=f(x')$$ but I dont know what happens with the $x_3$ component in the above integral.
I dont want a solution to the integral, I just would like to know how to perform this kind of integral, especially the component without a $\delta$.