The integral $\int\limits_0^\pi sin^3 \theta d \theta$ is given by

$I = \int\limits_0^\pi sin^3 \theta d \theta \\ = \int\limits_0^\pi \bigg(\frac{3sin\theta - sin 3 \theta}{4}\bigg)d\theta \hspace{2cm} sin 3 \theta = 3 sin \theta - 4 sin^3 \theta \\ = \bigg[\frac{-3}{4}cos \theta\bigg]_0^\pi = \bigg[\frac{\omega s 3 \theta}{12}\bigg]_0^\pi = \bigg[ \frac{3}{4}+ \frac{3}{4}\bigg]-\bigg[ \frac{1}{12}+ \frac{1}{12}\bigg] = \frac{4}{3}$