homework and exercises - Question about an integration by parts in Feynman's Quantum Mechanics

I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I saw it done once methodically it would clarify a lot. I have no problem with integration by parts in general.

On p. 27 he says that upon integration by parts the variation in $S$ becomes

$$ \delta S = \left[\delta x \frac{\partial L}{\partial \dot{x}} \right]_{t_a}^{t_b} - \int_{t_a}^{t_b}\delta x\left[\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}\right] dt. \tag{2-6} $$

Now

$$ S = \int_{t_a}^{t_b}L(\dot{x},x,t) dt \tag{2-1}$$

in which $L$ is the Lagrangian

$$ L = \frac{m}{2}\dot{x}^2 - V(x,t)\tag{2-2}$$

and he says that to a first order

$$\delta S = S[\bar{x}+ \delta x] - S[x] = 0. \tag{2-4}$$

He shows $S[x+\delta x]$ explicitly in (2-5) and from this derives (2-6).