I am trying to find the extremals for the following expression:
$$L(x, y(x), y'(x)) = y^2 + y'^2 -2x\sin(x)$$
Since that last term does not depend on either $y$ nor $y'$ it should not matter since it vanishes under the partial derivatives of the Euler Lagrange equation right?
Put otherwise the integral would be:
$$\int L dx = \int y^2 + y'^2 dx -2\int x\sin x(x)dx$$
Clearly the last term is a constant for a fixed choice of interval, thus anything minimizing the above would equivalently minimize
$$\int L dx = \int y^2 + y'^2 dx$$
Right?
Exactly. If $L'(x,y,y') = y^2 + (y')^2$, then the action functionals associated to $L$ and $L'$ differ by a constant (assuming, say, that the domain of all functions $y$ considered is a fixed interval $[a,b]$). Namely, the constant is $$-\int_a^b 2x\sin(x)\,{\rm d}x,$$and (variational) derivatives kill constants.