So I have a wave function in spherical polars: $$ \left(r\sin(\theta)e^{i\phi}\right)^l\text{exp}\left({\frac{-r}{(l+1)a}}\right) $$
and I have to show $E[r^k]$ is
$$ \frac{\int_0^\infty r^{2(l+1)+k}\text{exp}\left({\frac{-2r}{(l+1)a}}\right)}{\int_0^\infty r^{2(l+1)}\text{exp}\left({\frac{-2r}{(l+1)a}}\right)} $$
I know the formula for the expectation but cant seem to get the required form, any help will be appreciated!
Let $\displaystyle \psi(r,\theta,\phi)=(r\sin(\theta)e^{i\phi})^\ell\, e^{-r/((\ell+2)a)}$ be the wave function. Then, the expectation, $E[r^k]$, is given by
$$\begin{align} E[r^k]&=\frac{\int_0^{2\pi}\int_0^\pi \int_0^\infty r^k \left|\psi(r,\theta,\phi)\right|^2\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}{\int_0^{2\pi}\int_0^\pi \int_0^\infty \left|\psi(r,\theta,\phi)\right|^2\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}\\\\ &=\frac{\int_0^{2\pi}\int_0^\pi \int_0^\infty r^k \left|(r\sin(\theta)e^{i\phi})^\ell\, e^{-r/((\ell+2)a)}\right|^2\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}{\int_0^{2\pi}\int_0^\pi \int_0^\infty \left|(r\sin(\theta)e^{i\phi})^\ell\, e^{-r/((\ell+2)a)}\right|^2\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}\\\\ &=\frac{\int_0^{2\pi}\int_0^\pi \int_0^\infty r^k (r^{2\ell}\sin^{2\ell}(\theta)\, e^{-2r/((\ell+2)a)})\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}{\int_0^{2\pi}\int_0^\pi \int_0^\infty (r^{2\ell}\sin^{2\ell}(\theta)\, e^{-2r/((\ell+2)a)})\,r^2\sin(\theta)\,dr\,d\theta\,d\phi}\\\\ &=\frac{\int_0^\infty r^{2(\ell+1)+k}e^{-2r/((\ell+2)a)}\,dr}{\int_0^\infty r^{2(\ell+1)}e^{-2r/((\ell+2)a)}\,dr} \end{align}$$
as was to be shown!