How can the momentum of a wavefunction be determined using Fourier transforms?

[function22]

Homework Statement

Wavefunction is of form:
ψ(x) = e^ikx
Find momentum and energy of this state.

Homework Equations

Fourier transform of ψ(x) to get to momentum space
or is it
<p> = integral from -infinity to infinity of ψ* (h/i) * derivative wrt x of ψ dx

The Attempt at a Solution

I initially tried the second approach, but it didn't work, I got an infinite answer. Someone said to instead convert the function to momentum space, I used the Fourier transform but when I do that, my integral in the Fourier transform is -infinity to infinity of an oscillating function that doesn't decrease and is undefined.

I have no idea now how to proceed. I've worked on this question for hours, I searched the textbook, google, etc. and could not find anything useful.

 

[vela]

Hint: One representation of the Dirac delta function is
$$\delta(x-x_0) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{-ik(x-x_0)}\,dk$$

Another way you could approach the problem is to simply apply the momentum operator to that function and interpret what the equation means.