$$ f(t)=\frac{\sin(at)}{t} $$

Since the term is parameterized, it's easy to see that if I take the first derivative with respect to 'a', then the function becomes considerably easier. I do this to the Fourier Transform and obtain: $$ \frac{\partial }{\partial a}\Im (f(t))=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }\cos(at))e^{itx}dt $$

However, this is an integral of an even function times an odd function, which equals 0 and raises my suspicion. I've tried implementing Euler's cosine form and got nowhere.

Also I'm using the imaginary symbol as the Fourier transform. Why? It looks cool.