= Solution
On the outward-oriented <sphere> of radius $R$, $\mathbf n=\mathbf x/R$, so $\mathbf x\cdot d\mathbf S=R\,dS$. Hence
$$
\boxed{I=\frac1{R^2}\operatorname{Area}(S_R)=4\pi.}
$$
Away from the origin, the <divergence and curl of a radial vector field> with $f(r)=r^{-3}$ give
$$
\nabla\cdot\left(\frac{\mathbf x}{r^3}\right)=r(-3r^{-4})+3r^{-3}=0.
$$
If the enclosed volume excludes the origin and its boundary avoids it, this field is smooth throughout the volume. The <divergence theorem> then gives \b[$I=0$]. The nonzero <sphere> flux is compatible with zero <divergence> away from the origin because the enclosed singularity prevents applying that smooth-volume theorem directly to the full ball.
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