This is from Daniel Liberzon's book on Optimal Control, see section 2.5.2 in http://liberzon.csl.illinois.edu/teaching/cvoc.pdf.
Consider a basic calculus of variations problem: $$ J(y) = \int_a^b L(x,y,y')dx $$
where $y = y(x)$ and $L : \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is the Lagrangian. Now consider an equality constraint $$ M(x, y, y') = 0 \;\;\; \forall x \in [a,b] $$
Liberzon states that the Euler-Lagrange equation must hold for the augmented Lagrangian then: $$ L + \lambda^*(x)M $$
and this is equivalent to considering the minimization of $$ \int_a^b Ldx + \int_a^b \lambda^*(x)Mdx $$
Note that Liberzon misses out the asterisk for $\lambda^*(x)$ in the second integral, but I assume that is a typo. Albeit he is persistent with missing it in the next few sentences.
He then states that $\lambda$ no longer needs to be constant, since $M$ is identically 0.
My question is basically what's going on here? If $M=0$ $\forall x \in [a,b]$, then why do we even have the second integral? I understand the derivation of the augmented Euler-Lagrange equation in section 2.5.1 with the integral constraint and why the Lagrange multiplier is there, but in this case whatever we pick as $\lambda$, the second integral vanishes.
In fact, he goes on to compare this sum of integrals to equation (2.51) which leads to the same situation. Can anyone explain?
P.S. I know my question is a bit of a mess, so if anything is clear, do ask for clarification!