$\mathrm{F}=\int 2 \mathrm{dF} \cos \theta$

$=\int 2\left[\frac{\mu_{0}(\mathrm{d} i) i_{0}}{2 \pi \mathrm{R}}\right] \cos \theta$

$=\frac{\mu_{0} i_{0}}{\pi R} \int d i \cos \theta$

where,

$\mathrm{d} i=\frac{i}{\pi \mathrm{R}} \times \mathrm{R} \mathrm{d} \theta=\frac{i \mathrm{d} \theta}{\pi}$

$\therefore \,\,\, \mathrm{F}=\frac{\mu_{0} i_{0}}{\pi \mathrm{R}} \int \frac{(i \mathrm{d} \theta) \cos \theta}{\pi}$

$=\frac{\mu_{0} i_{0} i}{\pi^{2} \mathrm{R}} \int_{0}^{\pi / 2} \cos \theta \mathrm{d} \theta=\frac{\mu_{0} i_{0} i}{\pi^{2} \mathrm{R}}$