Vector potential of infinite wire and retarded potential

Eventhough there are some similar questions already, i have some trouble with the details of this task.

Given an infinite wire in z-direction (approx zero thickness) with the current $$I(t) =begin{cases} 0 & t le 0 \
I_0 & t gt 0
end{cases}$$

a) Calculate the vector potential $vec{A}$ in time-domain at the point $P(r=s, phi=0, z=0)$ in the far field.

b) Calculate the electric Field $vec{E}$ and the magnetic flux density $vec{B}$

I’m pretty sure i can solve part b) once the vector potential is calculated.

My approach would be: $$vec{A}(vec{r},t) = frac{mu_0}{4pi}int_V frac{vec{J}(vec{r}, t-|vec{r}-vec{r}’|/c)}{|vec{r}-vec{r}’|} mathrm{d}^3 r’ $$

Then i tried to use the far field approximation with the following result:
$$vec{A}(vec{r},t) = frac{mu_0}{4pi r}int_V vec{J}(vec{r}, t-(r-vec{e_r}cdotvec{r}’)/c) mathrm{d}^3 r’ $$

The point that i am stuck now is solving the integral. Given the current, is it even necessary to calculate the integral? Another point of trouble is the retarded potential and the $r$ dependency of the time in $J(vec{r},t)$.

In the task there is also a hint: $int frac{mathrm{d}x}{sqrt{x^2+z^2}}=ln(x + sqrt{x^2+z^2})$ which leads me to believe, that i don’t have to use polar coordinates to solve the integral. So i can probably replace $r$ with $x$ and $r’$ with $z$ which further simplifies things.

$$vec{A}(x,z,t) = frac{mu_0}{4pi x}int_V vec{J}(x, z, t-x/c) mathrm{d}V $$

Is this correct so far? I don’t really see how i can get to a point where i can apply the hint and solve the integral.

Maybe you can guide me in the right direction.

Edit:

Maybe the current density could be written as:
$$vec{J}(x,y,t) = vec{e_z}delta(x)delta(y)I(t)$$
I don’t see how i can directly use this. There is probably some mistake i’ve done ealier. Maybe already in the far field approximation?