TITLE: Is there something fundamental that makes integrals nontrivial to solve?
QUESTION [32 upvotes]: First of all, I apologize if this isn't the right forum.
The thing with derivatives is that once you learn product/quotient/chain rule and the formulas for trig/exponential/logarithmic functions, you can take the derivative of any function (as far as I can tell).
With integrals, the methods to solving are much more complex, with u-subs, integration by parts, partial fraction decomposition, etc. And then you have equations like $\int \sqrt{\sin x\cos x}dx$ which are simple to write but impossible to solve in terms of elementary functions. Does this have something to do with the fundamental theorem of calculus, and how the integral is defined as the area under a curve, or something else?
I'm only in Calc II right now, so it's possible that I will learn more about why this is later on in my math "schedule".

REPLY [6 votes]: This question interested me emough to make me do a bit of searching and link following. TL;DR: there is, and it's the Turing Halting problem!
Fiest I found this SO answer: How can you prove that a function has no closed form integral? , which is pretty much an expansion of what WB-man explains very simply above, but more formally. And then I found the answer, currently fourth down, pointing to the Risch Algorithm https://en.wikipedia.org/wiki/Risch_algorithm
So: there is an algorithm that may obtain an anti-derivative in finite time. But it's not a true algorithm in that it may fail to terminate (halt), and therefore cannot tell us anything about the possible non-existence of the closed-form anti-derivative.