While trying to find out the electric field due to a charged sphere without using Gauss Law, I encountered an integration $$\iiint\frac{r^2\sin\theta \,dr\,d\theta \,d\phi}{\left|\vec{r}-\vec{r'}\right|^3}\left(\vec{r}-\vec{r'}\right)$$ Here, $\vec{r'}$ is a constant, while $\vec{r}$ is the radial vector.
The exact part which is troubling me the most is $$\int_0^{r_o}\frac{r^2\left(\vec{r}-\vec{r'}\right)}{\left|\vec{r}-\vec{r'}\right|^3}dr$$ 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 $\cos\theta\cos\phi\hat{i}+\cos\theta\sin\phi\hat{j}+\sin\theta\hat{k}$, 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?