Integration of a vector function in spherical coordinates, where the unit vector cannot be represented as the standard radial unit vector?

While trying to find out the electric field due to a charged sphere without using Gauss Law, I encountered an integration $$iiintfrac{r^2sintheta drdtheta dphi}{left|vec{r}-vec{r’}right|^3}left(vec{r}-vec{r’}right)$$
Here, $vec{r’}$ is a constant, while $vec{r}$ is the radial vector.

The exact part which is troubling me the most is $$int_0^{r_o}frac{r^2left(vec{r}-vec{r’}right)}{left|vec{r}-vec{r’}right|^3}dr$$
or, $$int_0^{r_o}frac{r^2(widehat{vec{r}-vec{r’}})}{left|vec{r}-vec{r’}right|^2}dr$$
Here, the unit vector isn’t $hat{r}$, which can be written as $costhetacosphihat{i}+costhetasinphihat{j}+sinthetahat{k}$, as I saw in another answer Vector valued integral in spherical coordinates.
Rather, it is $(widehat{vec{r}-vec{r’}})$. This cannot be taken out of the integral as the unit vectors of Cartesian coordinates $hat{i}, hat{j}$ and $hat{k}$, since they are dependent on $theta$ and $phi$. So, how exactly should I proceed?

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I want to create a standard pbr material that is editable with the shader graph, but when I install the package and go the Create -> Shader tab, I only have 2 graph options: Empty Shader Graph, and Sub Graph. Isn’t there a shader graph that starts me off with a basic pbr material with color and reflections? I’m using unity 2020, and the default render pipeline. Do I need to switch to ldrp or hdrp?

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Let $X$ denote a random variable, with mean $mu$ and variance $sigma^2$. Suppose the following condition to be true, where

begin{equation}
P(X=mu+sigma)=P(X=mu-sigma)=0.5
end{equation}

Is there any specific shape or probability distribution that would satisfy this?

I believe this is saying that all possible values of $X$ lie within one standard deviation, if this is true, what else does it imply about $X$?

Thanks in advance for any clarifications.

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