Chapter 8: Techniques of Integration - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 462: 31

$$\sqrt{2}$$

We integrate as follows: \begin{align*} \int_{0}^{\pi / 2} \theta \sqrt{1-\cos 2 \theta} d \theta\\&=\int_{0}^{\pi / 2} \theta \sqrt{2}|\sin \theta| d \theta\\ &=\sqrt{2} \int_{0}^{\pi / 2} \theta \sin \theta d \theta\\ &=\sqrt{2}[-\theta \cos \theta+\sin \theta]\bigg|_{0}^{\pi / 2}\\ &=\sqrt{2} \end{align*}