First integral for a variational integrand independent of the function value ($f_v(x,u^{\prime }(x))=c$)

If a smooth functional has integrand $f(x,v)$ independent of the function value, a weak local minimizer satisfies the displayed first integral almost everywhere. Its first variation gives $\int f_v(x,u^{\prime }){\phi }^{\prime }=0$ for every compactly supported test function. Thus the distributional derivative of $f_v(x,u^{\prime })$ vanishes, making it constant on the interval. Equivalently, any smooth compactly supported function with integral zero is the derivative of such a test function, so the first-variation identity says that $f_v(x,u^{\prime })$ agrees with a single constant against all tests. This reduces the Euler-Lagrange equation from a second-order equation to an algebraic relation in $u^{\prime }$.