Some calculations when proving Ehrenfest Theorem 
Here  $\psi$ is a function of $x$ and $t$ and $\psi^*$ means the conjugate of $\psi$, and the integrals are done on $x$. On the the first term in the first integral on the right, I am supposed to do integration by parts twice and find that the whole integral cancels out, but so exasperatingly the integral never cancels out.... I have calculated more than $20$ times and just got some conjugates integrated together...
 Could anyone show me how to do the correction calculation?
 A: I'll start with where your computation left off
$$
\frac{d}{dt} \langle p_x \rangle \;\; =\;\; \int -\frac{\hslash^2}{2m}\left [ \frac{\partial^2\psi^*}{\partial x^2}\frac{\partial \psi}{\partial x} - \psi^* \frac{\partial^3 \psi}{\partial x^3} \right ]dx + \int \left[ U\psi^*\frac{\partial \psi }{\partial x} - \psi^* U \frac{\partial \psi}{\partial x} - \frac{\partial U}{\partial x} |\psi|^2 \right ]dx.
$$
For the second integral, two of the terms cancel out and we are left with $\int - \frac{\partial U}{\partial x} |\psi|^2dx$, which is the term we want to keep in the end since this matches the classical equation $\frac{dp}{dt} = -\frac{dV}{dx}$.  Why must the other integral go to zero?  Two applications of integrations by parts on the $\psi^* \frac{\partial^3 \psi}{\partial x^3}$ term will allow you to cancel it out with $\frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x}$ term.  The reason that this technique is able to work is because in quantum mechanics we assume that the functions $\psi$ are normalized and approach zero as we take the limit as $x \to \pm \infty$, this is why the evaluation terms go to zero when we do integration by parts in Ehrenfest's theorem.  Your final result should be 
$$
 \frac{d\langle p\rangle}{dt}  \;\; =\;\; \left \langle - \frac{\partial U}{\partial x} \right \rangle.
$$
Update
Hopefully my explanation for the second integral made sense.  I will compute the first integral.  For clarity, I will ignore the constant $-\frac{\hslash^2}{2m}$ in the front, since this won't affect the computation.
\begin{eqnarray*}
\int\frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} - \psi^* \frac{\partial^3 \psi}{\partial x^3} dx  & =& \int \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} dx - \int \psi^* \frac{\partial^3 \psi}{\partial x^3} dx \\
& = & \int \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} dx - \left [ \underbrace{\left .\psi^* \frac{\partial^2 \psi}{\partial x^2} \right |_{-\infty}^\infty}_{= \; 0} - \int \frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2} dx \right ] \\
& = & \int \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} dx + \int \frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2} dx \\
& = & \int \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} dx + \left [ \underbrace{\left . \frac{\partial \psi^*}{\partial x} \frac{\partial \psi}{\partial x} \right |_{-\infty}^\infty}_{= \; 0} - \int \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} dx \right ] \\
& = & \int \left (\frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} -  \frac{\partial^2 \psi^*}{\partial x^2} \frac{\partial \psi}{\partial x} \right ) dx\\
& = & 0.
\end{eqnarray*}
The two subtle points come with evaluating the integration by parts $\left . \psi^* \frac{\partial^2\psi}{\partial x^2} \right |_{-\infty}^\infty$ and $\left . \frac{\partial \psi^*}{\partial x} \frac{\partial \psi}{\partial x} \right |_{-\infty}^\infty$.  This is where we use the assumption that the state function satisfies $\lim_{x \to \pm \infty} \psi(x) = 0$ and $\lim_{x \to \pm \infty} \frac{\partial \psi}{\partial x} = 0$, and the same limits hold for their conjugates.  The reason for this is that we won't get a physically realizable distribution for our quantum system.
A: 
This is what I have thought so far and where I'm stuck at. Sorry for such a picture..the only available device is an iPad right now...
Now here, how should I progress to make the cancelling? I am just totally stuck.
