Compute the integral

by using Parseval's formula for Fourier transform

where means the Fourier transform of a function

Using Parseval's formula we can rewrite the integral as

with their inversions as follows

by the table 2 in Folland p.223

and the other one

which I calculated myslef as follows

we let and then the Fourier transform of g(x) is

then using formula in Folland again table 2

and applying it to our function

we get

going back to the integral and plugging in everything in the Parseval's formula gives me smth like this

in the solution given by our teacher we find

how did he get this on the left hand side and then on the right hand side??