Solution
= Solution
Insert the incident plane wave for $\psi$ in the <integral>. At large $r$, $|\mathbf r-\mathbf r'|\simeq r-\widehat{\mathbf r}\cdot\mathbf r'$, so the outgoing amplitude is
$$
f(\mathbf k',\mathbf k)=-\frac{m}{2\pi\hbar^2}\widetilde V(\mathbf q),
\quad
\widetilde V(\mathbf q)=\int e^{-i\mathbf q\cdot\mathbf r}V(\mathbf r)\,d^3r,
\quad\mathbf q=\mathbf k'-\mathbf k.
$$
\b[Thus <Born approximation> gives the stated differential cross section.]