It's not going to be useful to understand Lagrangians in detail unless you're actually doing the physics yourself -- especially because the whole framework is s

It's not going to be useful to understand Lagrangians in detail unless you're actually doing the physics yourself -- especially because the whole framework is super problematic, pedagogically (I hope it gets 'refactored' someday into something more intuitive). The important part is: a Lagrangian is a function that encodes the relationship between all of objects in a physical model, in such a way that you can 'add terms' to the function to add new interactions in your model. That much, at least, is quite elegant, and it's kinda miraculous that such a thing exists at all. For what it's worth, integration is really not that tricky. If I had to guess what really deterred you was the obsession with annoying proofs about dividing functions into little rectangles and summing them up and taking their limit. That stuff is totally unnecessary, imo. It's easier to think of integration as the inverse of differentiation. If differentiation tells you how fast a function is changing, then differentiating your position gives your velocity. Integration is the opposite: integrating your velocity over a length of time tells you how the position changes. That's a pretty intuitive idea: if I told you the velocity vector for the whole duration of a road trip, you can probably figure out where I ended up, if I also tell you where I started. That's all integration is.