= Solution
Let $s=|\mathbf r-\mathbf r'|$ and introduce the <magnetic vector potential> $\mathbf A(\mathbf r)=\mu_0(4\pi)^{-1}\int\mathbf J(\mathbf r')/s\,dV'$. Differentiation is with respect to $\mathbf r$, so $\mathbf J(\mathbf r')$ is constant under it. Hence
$$
\nabla\times\frac{\mathbf J(\mathbf r')}{s}=\nabla(s^{-1})\times\mathbf J(\mathbf r')=-\frac{\mathbf r-\mathbf r'}{s^3}\times\mathbf J(\mathbf r')=\frac{\mathbf J(\mathbf r')\times(\mathbf r-\mathbf r')}{s^3}.
$$
Taking the curl through the integral reproduces the <Biot-Savart law>, proving $\boxed{\mathbf B=\nabla\times\mathbf A}$ and the requested representation.
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