Problem: properties of Fourier Transforms in Quantum Mechanics

1. The problem statement, all variables and given/known data
I am trying to derive a property of the Fourier Transform of the wave function.
[tex]F[\psi(cx)]=\frac{1}{|c|}\overline{\psi}\left(\frac{p}{c}\right)[/tex]

2. Relevant equations
The Fourier transform of [tex]\overline{\psi}(p)[/tex] is defined as
[tex] \overline{\psi}(p)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{-\frac{i}{\hbar}px}\psi(x)dx[/tex]
and the inverse fourier transform
[tex]\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}px}\overline{\psi}(p)dp[/tex]

3. The attempt at a solution
I tried changing the integration variable
[tex]u=cx[/tex]
Then
[tex]\psi(u)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}pu}\overline{\psi}(p)dp[/tex]
But when I took the fourier transform and substituted [tex]u=cx[/tex] I did not manage to get the desired result.

 

Thanks, but why is there a modulus involved?