I need to calculate the (distributional) Fourier transform of $$ f(x) = \frac{x^2}{x^2+1}. $$
I unsuccessfully tried to find a differential equation for $f$ in order to solve the Fourier-transformed equation for $\hat{f}$. One may also could think of doing the integral through Residue calculus, but unfortunately the integrand doesn't converge.
Does anyone know how to do this exercise?
Subtracting $1$ (whose Fourier transform is Dirac delta, up to a normalization constant) gives $-1/(1+x^2)$, whose Fourier transform integral does converge, and can be evaluated by residues: a constant multiple of $e^{-|x|}$.