The average of inverse square radius $r^{-2}$ for any state quantum $|n\ell\rangle$ using the radial Schrödinger equation

Question

I have a problem understanding the solution provided for the problem here:

It's from the book "Quantum Mechanics - Concepts and Applications", 2nd ed., by N. Zettili (Wiley, 2009).

My issue is that I don't understand the second equality of eq. (6.223). Why do these two integrals cancel each other out?

Answer 1

Hints to show second equality of eq. (6.223):

1. Integrate by parts twice as suggested by Count Iblis in a comment.

2. Argue that boundary terms at $r=\infty$ in eq. (6.223) are zero.

3. Argue that $$U_{n\ell}(r) ~=~ O(r)\quad \text{for}\quad r\to 0.\tag{*}$$ Eq. (*) is more tricky, and shown in e.g. this, this & this Phys.SE posts.

4. Use eq. (*) to argue that boundary terms at $r=0$ in eq. (6.223) are zero.