The most general solution for the Schrödinger Equation with $V = 0$ :
$$\Psi(x,t)=\int_{-\infty}^{\infty} A(k) e^{ikx} e^{−i\hbar k^2t/2m} dk$$
To normalize this, what constraints will be placed on $A(k)$?
What is implied about $A(k)$ from:
$$\int_{-\infty}^{\infty}\left|\;\int_{-\infty}^{\infty} A(k) e^{ikx} e^{−i\hbar k^2t/2m} dk\;\right|^2\;dx = 1$$
The constraint is,
$$\int \psi_p(x,t)^* \psi_q(x,t) dx = \delta(p-q),$$
Where $\delta$ is the Dirac delta function.
You also need to understand how to properly square an integral.
$$ \Big| \int f(k) dk \Big|^2 = \Big( \int f(k) \ dk \Big)^* \Big( \int f(q) \ dq\Big) = \int \int f^*(k) f(q) \ dk dq$$