Answer
$x$ = $(2n+1){\pi}±\frac{\pi}{3}$, n an integer
Work Step by Step
$f(x)$ = $x+2sinx$ has a horizontal tangent when $f'(x)$ = $0$
$f'(x)$ = $1+2cosx$
$0$ = $1+2cosx$
$cosx$ = $-\frac{1}{2}$
$x$ = $\frac{2\pi}{3}+2{\pi}n$ or $\frac{4\pi}{3}+2{\pi}n$ where n is an integer
$x$ = $(2n+1){\pi}±\frac{\pi}{3}$, n an integer
