Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function
$$\frac{\partial f(x,t)}{\partial t} = - \frac{\partial }{\partial x}\left( D^{(1)}(x,t) f(x,t)\right) + \frac{\partial^2 }{\partial x^2}\left( D^{(2)}(x) f(x,t)\right) $$
I'm interested in calculating $\frac{\partial }{\partial t}\mathbb{E}[f]:= \frac{\partial }{\partial t}\int_{-\infty}^{\infty}xf(x,t)dx$.
I've come to the equation: $$\frac{\partial }{\partial t}\int_{-\infty}^{\infty}xf(x,t)dx = -\int_{-\infty}^{\infty} x\frac{\partial}{\partial x} \left( D^{(1)}(x) f(x,t)\right) + \int_{-\infty}^{\infty} x\frac{\partial^2}{\partial x^2} \left( D^{(2)}(x) f(x,t)\right)$$
By integration by parts, $$\frac{\partial }{\partial t}\int_{-\infty}^{\infty}xf(x,t)dx = - \left[ x D^{(1)}(x) f(x,t) \right]^{\infty}_{-\infty} + \int_{-\infty}^{\infty} D^{(1)}(x) f(x,t) dx + \left[ x \frac{\partial}{\partial x} \left( D^{(2)}(x) f(x,t) \right) \right]^{\infty}_{-\infty} - \int_{-\infty}^{\infty} \frac{\partial}{\partial x} \left( D^{(2)}(x) f(x,t) \right) dx$$
How can I simplify furthermore? And to what extent it can be simplified?