My task is to
1) compute the Fourier transform of the function
2) compute the integral
1) I can write my function as and by using the formula
which finally gives gives
which agrees with the answer.
2) this one is a bit tricker and somehow it seems like I am on the correct track except for the sign I get....
I use Plancherel formula for Fourier transform to solve this integral, namely
and we have
and we know from the part 1) that
then our integral will be
the answer should be where do I make mistake?