= Solution
The normalized initial <Gaussian wave packet> is real and even, so the integrands for $\langle x'\rangle_0$ and $\langle p'\rangle_0$ are odd. Its second <position> moment is the elementary <Gaussian integral> $\langle x'^2\rangle_0=1/(2\alpha)$. Since $\partial_x\Psi=-\alpha x\Psi$ and $\partial_x^2\Psi=(\alpha^2x^2-\alpha)\Psi$,
$$
\langle p'^2\rangle_0
=\hbar^2\left(\alpha-\alpha^2\langle x'^2\rangle_0\right)
=\frac{\hbar^2\alpha}{2}.
$$
Moreover $\langle x'p'\rangle_0=i\hbar\alpha\langle x'^2\rangle_0=i\hbar/2$ and $\langle p'x'\rangle_0=-i\hbar/2$. Thus
$$
\boxed{\langle x'\rangle_0=\langle p'\rangle_0
=\langle x'p'+p'x'\rangle_0=0,\qquad
\langle x'^2\rangle_0=\frac1{2\alpha},\quad
\langle p'^2\rangle_0=\frac{\hbar^2\alpha}{2}.}
$$
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