Evaluate the following integral over all space, where $\gamma>0$ and $r^2=x_px_p$ $$\int_V r^{−3}e^{−\gamma r^2} x_ix_j dV$$
I have quite a few questions here that I began to list, but I struggled to articulate myself, I also have questions at more or less every step and so I would ask simply for a full solution if possible.
I am able to post my thoughts if someone really thinks it would help, but honestly they're questions related mostly to what the question is actually asking rather than struggling with the arithmetic. I have not met a question like this before and so I'm struggling simply with things like not quite understanding what $x_px_p$ is exactly, what we would need to change $dV$ to and whether we need to evaluate with $i,j,(p?)$ taking each of the values $1,2,3$.
Again, I know these are stupid questions, but I have not met a question like this before and I have more to work on so I think seeing a full solution would help with the rest.
Thank you and any help is appreciated.
$$\begin{align} \int_V \frac{1}{r^3}e^{-\gamma r^2} x_ix_j\,dV&=\int_0^{2\pi}\int_0^\pi f_i(\theta,\phi)\,f_j(\theta,\phi)\left(\int_0^\infty \frac{1}{r^3}e^{-\gamma r^2}r^2\,r^2\,dr\right) \sin(\theta)\,d\theta\,d\phi\\\\ &=\frac{1}{2\gamma}\int_0^{2\pi}\int_0^\pi f_i(\theta,\phi)\,f_j(\theta,\phi) \sin(\theta)\,d\theta\,d\phi\\\\ \end{align}$$
where $f_1(\theta,\phi)=\sin(\theta)\cos(\phi)$, $f_2(\theta,\phi)=\sin(\theta)\sin(\phi)$, and $f_3(\theta,\phi)=\cos(\theta)$.
Note that if $i\ne j$, then the integral of interest vanishes due to the orthogonality of the set $\{1,\cos(\phi),\sin(\phi)\}$ for $\phi \in [0,2\pi]$.
If $i=j=1$, then
$$\int_0^{2\pi}\int_0^\pi f_i(\theta,\phi)\,f_j(\theta,\phi) \sin(\theta)\,d\theta\,d\phi=\int_0^{2\pi}\int_0^\pi \sin^3(\theta)\cos^2(\phi)\,d\theta\,d\phi=\frac{4\pi}{3} $$
If $i=j=2$, then
$$\int_0^{2\pi}\int_0^\pi f_i(\theta,\phi)\,f_j(\theta,\phi) \sin(\theta)\,d\theta\,d\phi=\int_0^{2\pi}\int_0^\pi \sin^3(\theta)\sin^2(\phi)\,d\theta\,d\phi=\frac{4\pi}{3} $$
If $i=j=3$, then
$$\int_0^{2\pi}\int_0^\pi f_i(\theta,\phi)\,f_j(\theta,\phi) \sin(\theta)\,d\theta\,d\phi=\int_0^{2\pi}\int_0^\pi \sin(\theta)\cos^2(\theta)\,d\theta\,d\phi=\frac{4\pi}{3} $$
Therefore, we have
$$\int_V \frac{1}{r^3}e^{-\gamma r^2} x_ix_j\,dV=\begin{cases}\frac{2\pi }{3\gamma}&,i=j\\\\0&,i\ne j\end{cases}$$