The potential created by an electron in hydrogen

[starvingautist]

Images aren't embedding properly, so forgive the urls.

This is the question I am attempting to answer:

https://imgur.com/a/mEcAl

and this is the start of my answer:

https://imgur.com/a/8MoDi

so.. I wasn't expecting just to get the coulomb potential at the end. This is supposed to be the potential caused by the shell electron. Have I done something wrong here? Misunderstood the question? Does the V_H in the question refer to something else?

 

[NFuller]

The issue is with how you simplified the square roots. There needs to be an absolute value sign.
$$\sqrt{r^{2}+r_{e}^{2}+2rr_{e}}-\sqrt{r^{2}+r_{e}^{2}-2rr_{e}}=r+r_{e}-|r-r_{e}|$$
This means that you have to break the radial integral into two pieces. One for $r>r_{e}$ and one for $r<r_{e}$.