I need to find the Fourier transformation of $f(x)=\frac{x}{(x^2+1)^2}$ by two different methods. one is using the property of Fourier transformation, another is computing the integral by definition.
For method 1 of using the property of Fourier transformation:
Since I already know the Fourier transformation of $g(x)=\frac{1}{x^2+1}$ is $\widehat g(\epsilon)=\pi e^{-\epsilon}$ and $\widehat{g(x)g(x)}=\frac{1}{2\pi}\widehat g *\widehat g(\epsilon)$. I can write $f=g(x)g(x)x.$ So I first compute $\widehat{g(x)g(x)}=\frac{1}{2\pi}\widehat g *\widehat g(\epsilon)=\frac{1}{2\pi}\int_y\widehat g(\epsilon -y)\widehat g(y)dy=\frac{1}{2\pi}\pi^2\int_ye^{-\epsilon +y}e^{-y}dy=\frac{1}{2\pi}\pi^2\int_ye^{-\epsilon }dy$
But I don't know how to calculate $\int_ye^{-\epsilon }dy$? Does $y$ go from $-\infty$ to $\infty$? I feel so confused.
Can anyone help me? Please. Thanks so much!