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

we let

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?