At time $t = 0$, a one-dimensional free wave packet for a particle of mass $m$ takes the form:

[![enter image description here][1]][1]


where $\alpha$ is a real constant. 
I need to find momentum amplitude $A(k)$ for this wave packet. And write the expression of the time-dependent wavefunction.

I know we can write a free wave packet as a linear combination of plane waves

[![enter image description here][2]][2] 


And we chose the amplitude for the different plane waves as a Gaussian

[![enter image description here][3]][3]

and further, I can use the fact that

[![enter image description here][4]][4]

But apart from these peices, I have no idea how to proceed.


  [1]: https://i.sstatic.net/NETBn.png
  [2]: https://i.sstatic.net/Yk0MG.png
  [3]: https://i.sstatic.net/9JLi8.png
  [4]: https://i.sstatic.net/vQumJ.png