$\sin x+\sqrt3\cos x=2\sin5x$ $\to 2\sin\left( x+\dfrac{\pi}{3}\right)=2\sin5x$ $\to \sin5x=\sin\left( x+\dfrac{\pi}{3}\right)$ $\to \left[ \begin{array}{l}5x=x+\dfrac{\pi}{3}+k2\pi\\ 5x=\pi-x-\dfrac{\pi}{3}+k2\pi\end{array} \right.$ $\to \left[ \begin{array}{l}x=\dfrac{\pi}{12}+\dfrac{k\pi}{2}\\x=\dfrac{\pi}{9}+\dfrac{k\pi}{3}\end{array} \right.$