You're fine up to and including this line:
To integrate, you can just eliminate the derivative symbol, because the LHS is a total derivative. That is,
At this point, I would solve for and integrate again.
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?
Again, everything looks good up until this point:
You've lost the cubic power! I also think you mean , not . Try working those corrections back through and see what you get.
Here's a little hint for doing math in LaTeX: copy and paste. Whenever you're simplifying things, just copy the equation from the previous line, and make whatever changes you want. You'll make fewer mistakes that way than if you re-type out the whole equation. I do this all the time, and it's not only a mistake-saver, it's a real time-saver.