= Solution
The <first Born approximation> substitutes the <incident wave> for the exact <wave> inside the scattering <integral>. Thus
$$
\boxed{R_{\mathrm B}=\frac{m^2}{k^2\hbar^4}\left|\int e^{2ikx}U(x)dx\right|^2.}
$$
For the attractive delta this is $\beta^2/k^2$, the leading small-$\beta/k$ expansion of the exact result. Physically one neglects the distortion of the incident state by repeated interaction with the potential. At low $k$ the Born result can exceed one, while the exact reflection remains bounded and approaches one; it is not valid in that limit.
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