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I do not understand a step here!

The integral is:

[tex]\delta S(x,t)=-mc \int_a^b u_i d \delta x^i =0[/tex]

and now they say one should do integration by parts, but I do not know how this should work here?

Where are my two functions?As far as I see there is only the four-velocity and I do not how it depends on the differential so I wanted to ask you whether one of you could explain this step to me.

The result should be:

[tex]\delta S(x,t)=-mc u_i \delta x^i |^b_a + mc\int_a^b \delta x^i \frac{du_i}{ds} ds[/tex]

I have not a clue what happened there! Can you help me?