What is the smallest $d$ such that $\int\cdots\int \frac{1}{(1+x_1^2 + \cdots + x_n^2)^d} \, dx_1\cdots dx_d$ converges? 
What power $d>0$ is the smallest integer such that in $\mathbb{R}^n$, 
  $$I(n) =\int_{-\infty}^\infty\int_{-\infty}^\infty \cdots\int_{-\infty}^\infty \frac{1}{(1+x_1^2 + \cdots + x_n^2)^d} \,dx_1\, dx_2 \cdots dx_n < +\infty$$
Hint: Think of the integral in "polar coordinates" where $r$ goes from $0$ to $\infty$, integrate over the sphere of radius $r$.

I have no idea how to start. More specifically, how do you do polar coordianates in $\mathbb{R}^n$? Some calculations show that if $n=1$ then $d=1$ would be good enough. But for $n=2$ and $n=3$, $d=2$ and the integrals would equal to $\pi^2$.

Think of $dx_1 \cdots dx_n$ as a volume. Given the the integrand has rotational symmetry, partition the space into spherical shells. The integrand is constant on the surface of this shell.
$$
I(n) = \int_0^\infty \frac{1}{(1+ r^2)^d} dV_r
$$
where $dV_r$ denotes the volume of the thin shell  spanning radius range $(r, r+dr)$. The volume of this thin shell scales as $r^{d-1} dr$ in radius. 
You should be able to take it from here.
