$Let = \alpha = k\omega $ ($k$ is constant)
$\frac{{d\omega }}{{dt}} = k\omega \,\,\,\,\,\,\,\,\,\left[ {also\,\frac{{d\theta }}{{dt}} = \omega \Rightarrow dt = \frac{{d\theta }}{\omega }} \right]$
$\therefore \frac{{\omega d\omega }}{{d\theta }} = k\omega \Rightarrow d\omega = kd\theta $
$Now\,\int\limits_\omega ^{\omega /2} {d\omega = k\int {d\theta } } $
$\int\limits_{\omega /2}^0 {d\omega = k\int\limits_0^\theta {d\theta \Rightarrow - \frac{\omega }{2} = k\theta \Rightarrow - \frac{\omega }{2} = K{\theta _1}} } $
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\theta _1} = 2\pi n} \right)$
$\therefore \theta = {\theta _1}\,\,or\,\,2\pi {n_1} = 2\pi n$
${n_1} = n$
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