Magnetic induction due to a rotating sphere

We consider a small ring making a polar angle θ with the equatorial plane.

\mathbf{di = \frac{\omega}{2\pi}\frac{\sigma (2\pi r^{2})\cos \theta d\theta}{4\pi r^{2}}=\frac{\omega \sigma}{4\pi}\cos \theta d\theta}

\mathbf{dB = \frac{\mu_{0}i\cos^{2}\theta}{2R}}

\mathbf{dB = \frac{\mu_{0}\omega\sigma \cos^{3}\theta d\theta}{8\pi R}}

B = \int^{\pi/2}_{-\pi/2}\mathbf{\frac{\mu_{0}\omega\sigma \cos^{3}\theta d\theta}{8\pi R}}

\mathbf{B = \frac{\mu_{0}\sigma \omega}{6\pi R}}

The answer is given as 23μ_oσωR.