How to find the Fourier transform when it is in the form of $\frac j\pi$ [$\frac{1}{(t-3)^2-1}$+$\frac{1}{(t+3)^2-1}$]

Question

So as the title suggests, I can compute fourier transforms in basic forms using the known properties. But how would one attempt to do this? How do I compute the integral? Or is there a pattern/pair that I could use? The equation is as follows x(t) = $\frac j\pi$ [$\frac{1}{(t-3)^2-1}$+$\frac{1}{(t+3)^2-1}$]

Answer 1

Hint: Use fractions decomposition then $$ \dfrac{j}{\pi}\left(\dfrac{1}{(t-3)^2-1}+\dfrac{1}{(t+3)^2-1}\right) = \dfrac{j}{2\pi}\left(\dfrac{1}{t-4}-\dfrac{1}{t-2}+\dfrac{1}{t+2}-\dfrac{1}{t+4}\right) $$ and apply Fourier Transform $${\cal F}\left(\dfrac{1}{t-a}\right)=\int_{-\infty}^{\infty}\dfrac{1}{t-a}e^{-jwt}dt=j\pi\operatorname{sgn}(w)e^{-jwa}$$ which the integral calculate with Cauchy principal value.