Brachistochrone problem with friction

[janonus]

Hey,
I am doing some research on the brachistochrone problem WITH frictions. I found the following demonstration on the web.
The beginning is ok. But I can't understand how the managed to pass between (29) and (30) and between (30) and ((32), (33)).

If someone could help me, it would be very nice !
thanks :)

The link to the article : http://mathworld.wolfram.com/BrachistochroneProblem.html

 

[Dr.D]

To go from 29 to 30, it is pretty clear that they have integrated one time. Eq(29) is second order while eq(30) is first order. What you need to do is convince yourself that 30 is indeed the result of integrating 29.

 

[janonus]

Yeah, I got it !
Tanks for your help Dr.D !

 

[physics_cosmos]

Hi, not sure if it is ok to revive an old question, I am also having trouble following some of the steps in this demonstration, in particular steps (28) to (29). I know what the Euler Lagrange equation is, but applying it to the integrand in (28) does not appear to simplify to (29). Would anyone be able to help me with this?