Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 1.16.
The fact that the normalization of the wave function is constant over time is actually a special case of a more general theorem, which is
for any two normalizable solutions to the Schrödinger equation (with the same potential). The proof of this follows a similar derivation to that in section 1.4 of Griffiths’s book.
The derivative in the integrand is (where we’re using a subscript or to denote a derivative with respect that variable):
From the Schrödinger equation
Inserting this into 1 and integrating gives zero because all wave functions go to zero at infinity. [Of course, the theorem doesn’t hold if and are solutions for different potentials, because in that case the potential term wouldn’t cancel out in 5.]