$\begin{aligned} & \int_0^{\pi / 2} \sin ^5 \frac{x}{2} \sin x \mathrm{~d} x \\ & =\int_0^{\pi / 2} \sin ^5 \frac{x}{2} \cdot 2 \sin \frac{x}{2} \cdot \cos \frac{x}{2} \mathrm{~d} x \\ & =2 \int_0^{\pi / 2} \sin ^6 \frac{x}{2} \cdot \cos \frac{x}{2} \cdot \mathrm{d} x \\ & \quad \frac{1}{\sqrt{2}} \\ & =4 \int_0^6 t^6 \mathrm{~d} t=\frac{4}{7}\left[t^7\right]_0^{\frac{1}{\sqrt{2}}}\left[\text { let } \sin \frac{x}{2}=t\right]\end{aligned}$
$=\frac{4}{7} \cdot\left(\frac{1}{\sqrt{2}}\right)^7=\frac{1}{14 \sqrt{2}}$