This question is related to $X\to\infty$ and $Y\to\infty$ independently.
Consider the following problem $\int_{-X}^{Y} \, \frac{2t}{t^2+1}$ where (i) where $X,Y\to \infty$ independently. (ii) where $X,Y\to \infty$ dependently
For (i), $\int_{-X}^{Y} \, \frac{2t}{t^2+1}dt=\ln |Y^2+1|-\ln |X^2+1|$ , since $X,Y\to \infty$ independently. So, $\int_{-X}^{Y} \, \frac{2t}{t^2+1}dt$ is divergent as $X,Y\to \infty$ independently.
For (ii), I do not know what I suppose to do. If $X=Y$ or $X^2=Y^2$, or some similar cases, the $\int_{-X}^{Y} \, \frac{2t}{t^2+1}dt$ would be convergent. But for other cases, for example, $X=Y^4$ would not be.
Any idea? Thank you in advance.
$\ln [Y^{2}+1]-\ln [X^{2}+1]=\ln \frac {Y^{2}+1} {X^{2}+1}=\frac {1+Y^{-2}} {(X/Y)^{2}+Y^{-2}}$. Note that $Y^{-2} \to 0$. So $\ln [Y^{2}+1]-\ln [X^{2}+1]$ has finite limit if and only if $\frac {1+Y^{-2}} {(X/Y)^{2}+Y^{-2}}$ has a finite positive limit if and only if $(X/Y)^{2}$ has a non-zero limit. Equivalently, $\frac X Y$ has non-zero limit.