For the following variational problem I have been told to show the Euler-Lagrange equation is identically zero.
$$L[u]:= \int_a^b \sin(u)u_x\,\mathrm dx $$
I have found it to be
$u_x\cos(u)-\sin(u)u_{xx}=0.$
Is this correct? And if so, does this always equal $0$?