For $V\in C_c^\infty(\mathbb R^3;\mathbb R)$, the formal first-order term in the Born expansion of the outgoing scattering amplitude is
\begin{align*}
f_{\mathrm{Born}}(k',k)
= -\frac{m}{2\pi\hbar^2}\int_{\mathbb R^3} e^{-i(k'-k)\cdot x}V(x)\,d\mathcal L^3(x),
\end{align*}
for $|k'|=|k|$.