When you make a linear superposition like this $\psi(\boldsymbol{r},t)=h^{-\frac{2}{3}}\int [C^{+}(\boldsymbol{p})e^{-i\omega t}+C^{-}(\boldsymbol{p})e^{i\omega t}]e^{(i\boldsymbol{p}\cdot\boldsymbol{r}/\hbar)}\, d^{3}\boldsymbol{p}$ where $\omega=\frac{1}{\hbar}(c^{2}p^{2}+m^{2}c^{4})^{\frac{1}{2}}$, the exponential factors are transparent to the integration?. And if I calculate the expectation value of the momentum gradient operator like this $<\hat{\boldsymbol{\nabla_{p}}}>=\int\int\,d^{3}\boldsymbol{p}\,d^{3}\boldsymbol{q}\,(\phi(\boldsymbol{q},t)\hat{\boldsymbol{\nabla_{p}}}\phi(\boldsymbol{p},t))$ where $\phi(\boldsymbol{p},t)=C^{+}(\boldsymbol{p})e^{-i\omega t}+C^{-}(\boldsymbol{p})e^{i\omega t}$ then, the gradient operator acts on the exponentials, being $\omega$ a function of $p$?

Closed as per community consensus as the post is
Not graduate-level upwards