This is a multivariable calculus problem from a past prelim exam. I have an answer for this written up (posted below), but it seemed rather time-intensive. If there is a slicker way to approach this problem, I'd appreciate seeing it. Thanks!
Recall that for a smooth function $f: \mathbb{R}^3 \to \mathbb{R}$, the Laplacian of $f$ is defined by $ \Delta f = \nabla \cdot ( \nabla f). $
Suppose that $f: \mathbb{R}^3 \to \mathbb{R}$ is a smooth function satisfying $f(\vec{x}) = 1/\|\vec{x}\|$ for $\|\vec{x}\| \geq 1$.
Verify that $\Delta f(\vec{x}) = 0$ for $\|\vec{x}\| \geq 1$.
Compute $\int_{\mathbb{R}^3} \Delta f \, dV$.