= Solution
The center-of-mass and relative terms in the Hamiltonian are independent, so the phase-space integral factorizes as stated. Gaussian integration gives
$$
\boxed{Z_{\rm trans}=\frac{V}{(2\pi\hbar)^3}
\left(\frac{2\pi M}{\beta}\right)^{3/2}}.
$$
In the large-volume limit the relative coordinate may be integrated over all space. Since
$$
4\pi\int_0^\infty r^2e^{-\beta\alpha r}dr
=\frac{8\pi}{(\beta\alpha)^3},
$$
we obtain
$$
\boxed{Z_{\rm int}=\frac1{(2\pi\hbar)^3}
\left(\frac{2\pi\mu}{\beta}\right)^{3/2}
\frac{8\pi}{(\beta\alpha)^3}}.
$$
The radial probability is proportional to $r^2e^{-\beta\alpha r}$, hence
$$
\boxed{\langle r\rangle=\frac{3}{\beta\alpha}=\frac{3k_BT}{\alpha}}.
$$
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