I got some help with some worked examples here (thanks Ackbeet).
I'm now trying a question from my text to see if I understand it correctly. Do I have the right idea?
Find the extremal for the following:
Then Euler-Lagrange eqn is
Integrating we get
where C is a constant.
Now - Integrate with respect to and we get
Using and we have
Solving these we get
so the extremal is . Is this right or did I stuff something?