Show that the function

\[G: \mathbb{R} ightarrow \mathbb{R}, \quad G(x):=\int_{\mathbb{R} \backslash\{0\}} \frac{\sin (t x)}{t\left(1+t^{2}ight)} d t\]

is differentiable and find \(G(0)\) and \(G^{\prime}(0)\). Use a limit argument, integration by parts for \(\int_{(-n, n)} \ldots d t\) and the formula \(t \frac{\partial}{\partial t} \sin (t x)=x \frac{\partial}{\partial x} \sin (t x)\) to show that

\[x G^{\prime}(x)=\int_{\mathbb{R}} \frac{2 t \sin (t x)}{\left(1+t^{2}ight)^{2}} d t\]