Answer
The integral diverges.
Work Step by Step
We have
$\int_{-\infty}^{\infty} \dfrac{ x dx}{1+x^2} \ dx=\lim\limits_{R \to \infty} \int_{-\infty}^0\dfrac{ x dx}{1+x^2} \ dx \\=\lim\limits_{R \to \infty} \dfrac{1}{2} \ln (R^2+1) \\=\infty$
Also, $\int_0^\infty \dfrac{x dx}{1+x^2}=\lim\limits_{R \to \infty}
\int_0^{R} \dfrac{x dx}{1+x^2}\\=\lim\limits_{R \to \infty} \dfrac{1}{2} \ln (R^2+1) \\=\infty$
Hence, the integral diverges.
