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Delta function/fourier transform

1. The problem statement, all variables and given/known data
Hello. This question is about Fourier transforms and the Delta function.

Find the fourier transform of:


2. Relevant equations



3. The attempt at a solution

I began to solve this problem and quickly got to a point where I saw the above mentioned delta function representation. Using Euler's formula, I re-wrote the sine term. My question is, can I use the delta function here:


Now that I have "said that outloud" it seems like the answer is of course not. I guess really I am just looking for a short cut to get out of doing a lot of math. Is there an efficient way of evaluating this integral without "trudging" through it?

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