I usually prefer to do this kind of problem by just pluggin' n' chuggin'.
Refer to the attachment below.
In lines 1 and 2 I have simply listed the FT of the wavefunction and its complex conjugate. On line 3 I have plugged these expressions into the k integral. It looks like a real mess this way, but after doing a bit of rearranging (I'm not going to bother with the Mathematical justification to do it) and you get line 4. Note the delta function integral at the end. I leave the rest to you.
I don't know what the rest of your course is going to require, but this sort of derivation crops up in a number of useful situations.
Dan
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