A current $I$ flows in an infinitely long wire with cross section in the form of a semicircular ring of radius $R$. The magnitude of the magnetic induction along its axis is:

Consider a small element of the semicircular ring subtending angle $d\theta$.The current in this element is $di=\frac{I}{\pi R}Rd\theta =\frac{I}{\pi }d\theta$.The magnetic field due to this element (infinitely long strip) is $dB=\frac{{\mu }_{0}di}{2\pi R}=\frac{{\mu }_{0}I}{2{\pi }^{2}R}d\theta$.By symmetry, the horizontal components cancel. The vertical component is:
$$B={\int }_0^{\pi }dB\sin \theta ={\int }_0^{\pi }\frac{{\mu }_{0}I}{2{\pi }^{2}R}\sin \theta d\theta$$
$$B=\frac{{\mu }_{0}I}{2{\pi }^{2}R}[-\cos \theta {]}_0^{\pi }=\frac{{\mu }_{0}I}{2{\pi }^{2}R}(1-(-1))=\frac{2{\mu }_{0}I}{{\pi }^{2}R}$$