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**Opalg** You have indicated the integral and the integrand correctly. The integral is not a function of x because it is a definite integral, which means that the x is a dummy variable that does not feature in the result of the integral. To take a simple example, $\displaystyle \int_0^1x\,dx = \frac12$. The integral there is not a function of x because it is just a constant, 1/2. When you integrate a function of two variables, such as $\displaystyle \int_{-\infty}^{\infty}|\psi(x,t)|^2\,dx$, the integral is a function of t, but because it is a definite integral, it is not a function of x.

There is a theorem which says that you can push the operation of differentiation with respect to t past the integral sign. But when you do so, it changes from an ordinary differentiation to a partial differentiation.