I will begin by saying that I don't want to dissuade anyone who doesn't know PDE from helping, so if you're just here for the integral, you can skip down to "Where I'm Stuck" (For future reference: $\vec{x}=(x,y,z)$, $\vec{x}_0=(x_0,y_0,z_0)$, $\vec{x}+\vec{x}_0=(x+x_0,y+y_0,z+z_0)$)
Integral set-up
In my PDE class, we were asked to solve the homogeneous wave equation ($U_{tt}-c^2\nabla^2 U=0$) for $U(\vec{x},t)$ on $\mathbb{R}^3$ where at $U(\vec{x},0)=xy^3z^2$ and $U_t(\vec{x},0)=z^2$. The general way to solve the problem is to split the problem into 2 parts:
a) Solve $V_{tt}-c^2\nabla^2 V=0$ where $V(\vec{x},0)=xy^3z^2$ and $V_t(\vec{x},0)=0$
b) Solve $W_{tt}-c^2\nabla^2 W=0$ where $W(\vec{x},0)=0$ and $W_t(\vec{x},0)=z^2$
So then $U=V+W$.
In class we derived a supposedly "simple" formula to solve part a ($\psi(\vec{x})=V(\vec{x},0)=xy^3z^2$): $$V=\frac{\partial}{\partial t}\Bigg (\frac{1}{4\pi c^2t}\iint_{|\vec{x}_0|=ct}\psi(\vec{x}+\vec{x}_0)\,dS_{\vec{x}_0}\Bigg )$$ I'm going to ignore the partial derivative and the coefficient later on in the question because calculation of that is trivial to the problem I am having.
Where I'm stuck
I need to evaluate the following integral: $$\iint_{|\vec{x}_0|=ct}(x+x_0)(y+y_0)^3(z+z_0)^2\,dS_{\vec{x}_0}$$ The $|\vec{x}_0|=ct$ and $\,dS_{\vec{x}_0}$ mean that we're essentially treating $x$, $y$, and $z$ as numbers (instead of variables) and integrating only by the variables with 0 subscript ($x_0$, $y_0$, and $z_0$) over a sphere of radius $ct$ (since theoretically this is for light waves). I'm not sure how I'm supposed to evaluate this, asI may have forgotten some of my calculus, but my confusion arises from the fact that typically when dealing with surface integrals, the integrand is a vector field (the result is the flux through an area). Although the integral now doesn't make physical sense to me, I can still try to work it out. Since I'm dealing with a sphere, I was thinking of doing the bounds of integration in spherical coordinates about a sphere of radius $\rho=ct$: $$\int_0^\pi\int_0^{2\pi}(x+x_0)(y+y_0)^3(z+z_0)^2\rho^2\sin{\phi}\,d\theta\,d\phi$$ Since $\rho^2=c^2t^2$ has no relation to $\theta$ and $\phi$, I can pull it out of the integral, leaving me with: $$c^2t^2\int_0^\pi\int_0^{2\pi}(x+x_0)(y+y_0)^3(z+z_0)^2\sin{\phi}\,d\theta\,d\phi$$ My thought from here is to expand my integrand, separate terms that don't involve any 0-subscripted variables, then use the rectangular to spherical coordinate conversions ($x_0=\rho\sin{\phi}\cos{\theta}$, $y_0=\rho\sin{\phi}\sin{\theta}$, $z_0=\rho\cos{\phi}$), then integrate manually. Is there an easier way to do this since the expansion will have quite a few terms? Also, what will I do with the new $\rho$'s that appear? Will they all simply be equal to $ct$ since I'm only looking at my integrand on the sphere itself? It is also worth noting that in class, it was pointed out that if I have terms of odd degree (with respect to $x_0$, $y_0$, and $z_0$) in my integrand, integrating those terms over a sphere will result in 0 due to symmetry, so that's why I'm leaning towards this path. Any other suggested methods would be helpful as I will most definitely have to be integrating similar integrals in the near future.