I am not 100% sure why we go from derivatives involving time to derivatives involving position. I will nevertheless thake the rightmost term in (1.29) and start from there. Let us ignore the constants.
\begin{equation} \int x\frac{\partial}{\partial x}\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)dx \end{equation} let us use integration by parts where $x = U$ and $\frac{\partial}{\partial x^\prime}\Big( \psi^{}\frac{\partial\psi}{\partial x^\prime} - \frac{\partial\psi}{\partial x^\prime}\psi^{} \Big)dx = dV$. Integrtion by parts says that
$$ \int U dV = UV\Big|_{-\infty}^{\infty} -\int V dU $$ where we are integrating over the entire real line.
Notice that $dU = dx$ and that $V = \int\frac{\partial}{\partial x^\prime}\Big( \psi^{}\frac{\partial\psi}{\partial x^\prime} - \frac{\partial\psi}{\partial x^\prime}\psi^{} \Big)dx =\Big( \psi^{}\frac{\partial\psi}{\partial x^\prime} - \frac{\partial\psi}{\partial x^\prime}\psi^{} \Big) $ Ignore the constant of integration. 955
$$\int x\frac{\partial}{\partial x}\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)dx = x\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)\Big|_{-\infty}^{\infty}- \int VdU = $$
$$ x\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)\Big|{-\infty}^{\infty}- \int VdU = $$ $$ x\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)\Big|{-\infty}^{\infty}- \int \Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big) dx $$
the sneaky assumption being made is that the wave fucntion $\psi$, its derivative and ints comple conjugate all go to zero at infinity faster than the linear term $x$ blows up. If you use a gaussian for $\psi$ you will see that this is so. Therefore the boundary term is just zero, i.e. $$x\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)\Big|_{-\infty}^{\infty} = 0 $$
hence $$\int x\frac{\partial}{\partial x}\Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big)dx = - \int \Big( \psi^{}\frac{\partial\psi}{\partial x} - \frac{\partial\psi}{\partial x}\psi^{} \Big) dx.$$
This should do it. I was not too careful with the integration dummy variables and a few other details but the framework of what you are trying to show is presented here. 955