Use the general relation
$$
\theta=\pi-2 \int_{r_{m i n}}^{\inf } \frac{\left(b / r^{2}\right) d r}{\sqrt{1-(b / r)^{2}-U / E}}
$$
To denive the Rutherford dispersion formula
$$
\frac{d \sigma}{d \Omega}=\left(\frac{k q Q}{4 E \sin ^{2}(\theta / 2)}\right)^{2}
$$