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:

Using linear current density per unit angle: $$\frac{dI}{d\theta} = \frac{I}{\pi}$$

Using magnetic field of an infinitely long straight filament at distance $$R$$:

$$dB = \frac{\mu_0 dI}{2\pi R} = \frac{\mu_0 I}{2\pi^2 R} d\theta$$

Resolving components symmetrically about the central axis:

$$B = \int_{-\pi/2}^{\pi/2} dB \cos\theta = \int_{-\pi/2}^{\pi/2} \frac{\mu_0 I}{2\pi^2 R} \cos\theta \, d\theta$$

$$B = \frac{\mu_0 I}{2\pi^2 R} [\sin\theta]_{-\pi/2}^{\pi/2} = \frac{\mu_0 I}{2\pi^2 R} (1 - (-1)) = \frac{\mu_0 I}{\pi^2 R}$$