Solving derivatives and integrals that you've not seen before I hope this is a good place to ask this as its not a direct math problem.
I've learnt derivatives and integrals for my calculus classes, but I am not overly happy with how these are taught when they become much more complex and it seems to be the case for a lot of others who study calculus at early university.
A lot of the time, we're given a list of "standard integrals/derivatives" that we can then solve what ever is thrown at us.
What we are not taught is how to solve an integral or derivative when you are given some thing that does not resemble any standard one in the list and cannot be manipulated to resemble a standard one given either.
So then how did early mathematicians actually solve them when they had no list of standard ones to refer to - are there methods you can take to try to find the solution and how do you then verify the solution is the correct one?
 A: Differentiation is purely mechanical — you apply one of a few dozen standard formulas, and you’re done.
Integration is different. You’re again given a few dozen standard formulas, but it takes skill to massage the problem into one for which the standard formulas are applicable. There are some standard tricks like partial fraction expansions and the Weierstrass substitution, and others described here, but you still need to be clever and persistent. It takes a lot of practice to be able to recognize the problem patterns and map them to the available solutions. I don’t know of any magic shortcuts, but there’s a pretty long list of semi-magical tricks here.
I spent years getting good at integration, and in some ways I wish I hadn’t. Software like Mathematica and Maple can do the job far better than I can; better than the vast majority of people, actually. I wish I’d spent the time studying Chinese or learning to play guitar, instead. Integration is an amusing intellectual exercise, but I’m not convinced that people need to be good at it, nowadays. I expect your calculus teachers would disagree with me, though, and they’re the ones making the rules that you have to live by.
If you’re trying to find definite integrals in real life (i.e. outside a calculus class), then you’ll probably need to use numerical methods. Many of the functions that arise in science and engineering don’t have nice tidy anti-derivatives.
