Fix $y\in \mathbb R.$
Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$
My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value of $I(y);$ is it true that, $\int_{\mathbb R} |I(y)| dy <\infty$ ?
The change of variables $x\leftarrow y+t$ allows us to conclude that the integral converges if and only if $n>3$. Moreover, it shows that $$I(y)=2\int_0^\infty\frac{1+t^2}{(1+t)^n}dt+2y^2\int_0^\infty\frac{dt}{(1+t)^n}$$ This can be calculated explicitly: $$ I(y)=2\left(\frac{1}{n-3}-\frac{1}{n-2}+\frac{1}{n-1}\right)+\frac{2y^2}{n-1} $$ and clearly, $\int_{\mathbb{R}}I(y)dy=+\infty$ in this case