I am unsure if this belongs in calculus, differential equation, advance applied math, etc.

So I would like to figure out the trick to

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Yes, I know this is an Inverse Fourier Transform, and I know there are tables. What I am looking for is a good explanation of how

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Also, I figure this might help me understand the Dirac-delta function a bit better, because I am unsure why

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Thank you in advance.