Electron boosted by an electric field

I have an electric field,

[itex]\vec{E}=E_0 e^{(x_1-ct)^2}\vec{e_2}[/itex]

that boosts an electron that initially is at rest at (0,0,0). I have to calculate the motion of the electron (supposing v<<c and Newton Law is valid). Then, calculate total flux of radiation.

As v<<c I suppose I can assume that the induced magnetic field due to the time-dependent electric field is negligible. Using Newton equation:

[itex]ma=q(E+E_{rad})[/itex]

In the text they give me the expression of the radiation electric field,

[itex]E_{rad}(\vec{x})=-∫d^3x' \frac{\dot{[\vec{j}]}}{c^2 |\vec{x}-\vec{x'}|}[/itex]

But I don't know how to calculate this.

My other doubt is about the calculation of the total flux, what expression shoud I use?

Thank you!