Single variable improper integral Say I have an integral of $x/(1+x^2)$ that goes from negative infinity to infinity, and then part it into two integrals $A + B$ (let $A + B = I_\text{tot}$) where $A$ and $B$ have the limits from R to Infinity and negative infinity to R, respectively, and where R is any R-number. Now, if I integrate e.g. A I get 0.5*(natural log of u) after u-substitution and A having the limit from R to Infinity we know that the integral (and even I_tot because infinity+B=infinity) will diverge, and now I want to know if it is correct to say "The integral diverges or The integral diverges to infinity"? Because I compared many improper integrals that diverged but I couldn't find any patterns to confirm any difference.

$$
\int_{-\infty}^r \frac x {1+x^2} \,dx = -\infty \text{ and }\int_r^\infty \frac x {1+x^2}\,dx = +\infty.
$$
One can say that one of these diverges to $-\infty$ and the other diverges to $+\infty$.  However, notice also that
$$
\lim_{r\to\infty} \int_{-r}^r \frac x {1+x^2}\,dx = 0\text{ and } \lim_{r\to\infty} \int_{-r}^{2r} \frac x {1+x^2}\,dx = \log_e 2>0.
$$
The fact that the rearrangement in the last line can yield two different numbers is something that can happen only if the positive and negative parts both diverge to infinity.
