ordinary differential equations - Extremising a functional with boundary conditions (Euler-Lagrange)

I need to determine all functions $u(x)$ that extremise the functional: $$I[u]= \int_{-\infty}^\infty \left[\frac{(u')^2}{2}+(1-\cos u)\right] \, dx$$
subject to the boundary conditions

$$\lim_{x \to -\infty} u(x)=0$$ and $$\lim_{x \to \infty} u(x) = 2\pi$$
I used the standard approach for finding the stationary points of a functional; that is, attempting to solve the Euler-Lagrange equation, but assuming I've attempted this correctly I arrive at
$$\frac{d^2u}{dx^2} = \sin u$$
which I believe is not (easily) directly solvable, so I'm presuming there's either another way to approach this problem or I've messed up somewhere. A point in the right direction would be great, thanks in advance