Application of Lagrange Multipliers in action principle

In Goldstein’s Classical Mechanics, he suggests the use of Lagrange Multipliers to introduce certain types of non-holonomic and holonomic contraints into our action. The method he suggests is to define a modified Lagrangian $$L^{‘}(dot{q},q;t) = L(dot{q},q;t) + sum^{m}_{i = 1}lambda_{i} f_{i}(dot{q},q;t),$$ where $f_{i}(dot{q},q;t)$ are $m$ equations of constraint, and $L$ the original Lagrangian. He then proceeds to define the action $S^{‘} = int_{t_{1}}^{t_{2}}L^{‘},dt$ and takes the variation of $S^{‘}$ to be zero, thus applying Hamilton’s principle.

My confusion in this approach arises from the way in which the Lagrange Multipliers are introduced. I don’t see why $sum^{m}_{i = 1}lambda_{i} f_{i}(dot{q},q;t)$ should be introduced inside the integral.

In multivariable calculus, the Lagrange multiplier system stems from the idea that if we want to extremize a function subject to certain constraints, then the gradient of the function will be proportional to a linear combination of the gradient of the constraint equations. Here, the function in question is the action, not the Lagrangian. So, I feel like the resolution should be that $$delta S + delta sum^{m}_{i = 1}lambda_{i} f_{i}(dot{q},q;t) = 0;, S = int_{t_{1}}^{t_{2}}L,dt$$ and not $$delta S^{‘} = 0; , S = int_{t_{1}}^{t_{2}}L^{‘},dt.$$

To me, it isn’t clear if this makes sense or if the two methods are equivalent.