Answer
$-\displaystyle \frac{4}{3}(\sin\theta+10)\sqrt{5-\sin\theta}+C$
Work Step by Step
$\sin 2\theta=2\sin\theta\cos\theta$
$\displaystyle \int\frac{\sin 2\theta}{\sqrt{5-\sin\theta}}d\theta=\int\frac{2\sin\theta\cos\theta}{\sqrt{5-\sin\theta}}d\theta=\quad$substitute $\left[\begin{array}{l}
u=\sin\theta\\
du=\cos\theta d\theta
\end{array}\right]$
$=2\displaystyle \int\frac{u}{\sqrt{5-u}}du$
Table of integrals:
$\color{blue}{55. \quad \displaystyle \int\frac{udu}{\sqrt{a+bu}}=\frac{2}{3b^{2}}(bu-2\mathrm{a})\sqrt{\mathrm{a}+bu}+C }$
$=2\cdot \displaystyle \frac{2}{3(-1)^{2}}[-1u-2(5)]\sqrt{5-u}+C$
$=\displaystyle \frac{4}{3}(-u-10)\sqrt{5-u}+C$
...bring $\theta$ back ($ u=\sin\theta$)...
$=-\displaystyle \frac{4}{3}(\sin\theta+10)\sqrt{5-\sin\theta}+C$
