You can't conclude that the integral is zero. What you can conclude though is that the Cauchy principal value is zero. The integral is defined as below. $\int_{-\infty}^{\infty} \frac{2x}{1+x^2} dx = \lim_{R_1 \rightarrow - \infty, R_2 \rightarrow \infty} \int_{R_1}^{R_2} \frac{2x}{1+x^2} dx = \lim_{R_1 \rightarrow - \infty} \int_{R_1}^c \frac{2x}{1+x^2} dx + \lim_{R_2 \rightarrow \infty} \int_{c}^{R_2} \frac{2x}{1+x^2} dx $ where $c \in \mathbb{R}$.
Unless both the integrals are finite, you cannot make sense of the integral since you will get something "like $(-\infty) + (+ \infty$)"
However note that if $R_1 \rightarrow -\infty$ and $R_2 \rightarrow \infty$ at the same pace i.e. $R_2 = -R_1 = R$, then it is called the Cauchy principal value and is zero.