I'm using this resource along with Griffith's Introduction to Quantum Mechanics to try and reproduce the Ehrenfest theorem.
From equation $(176)$ in the link above, we have:
$$\frac{d\langle p\rangle}{dt}=\int_{-\infty}^{\infty}\left[\frac{-\hbar^2}{2m}\frac{\partial}{\partial x} \left( \frac{\partial \Psi^*}{\partial x} \frac{\partial \Psi}{\partial x} \right) +V(x)\frac{\partial|\Psi^2|}{\partial x} \right] dx$$
I am able to get to here without issues, but next we have to show that:
$$\int_{-\infty}^{\infty}\left[\frac{-\hbar^2}{2m}\frac{\partial}{\partial x} \left( \frac{\partial \Psi^*}{\partial x} \frac{\partial \Psi}{\partial x} \right) \right] dx = 0$$
Which would only be true if:
$$\left. \frac{\partial \Psi}{\partial x} \right|^{x=\infty}_{x=-\infty} = \left. \frac{\partial \Psi^*}{\partial x} \right|^{x=\infty}_{x=-\infty} = 0$$
Is there a way to know this generally? It's obviously true in certain cases of the wave function (e.g. $\Psi(x)=\exp[-x^2]$). In general, I thought the only condition for normalization was that:
$$\left. \Psi \right|^{x=\infty}_{x=-\infty} = \left. \Psi^* \right|^{x=\infty}_{x=-\infty} = 0$$