Subtraction isn’t much harder than addition. Division isn’t much harder than multiplication. Logarithmization isn’t much harder than exponentiation. Then, what fundamentally makes integration so much more difficult than differentiation?
In: Mathematics
All of those other operations are simple mappings between numbers that more or less define their own inverses. Subtraction is addition with a negative number, so it literally is addition as soon as you get negative numbers involved. Division literally is multiplication once you get rationals involved, and both are addition on log scales e.g. on a slide rule where you can read off either the addition/multiplication or the corresponding subtraction/diffusion. Exponentiation is a monotonic function with one input, and we happen to know the series expansion for the inverse, so we gave it a name and now it’s a thing.
Differentiation is a mapping of functions to functions. There are very few functions that we have a name for and that we understand well. Mechanically, integration is sometimes easier than differentiation. If you have a function that’s continuous, it might not be differentiable but it will always (I think?) be integrable. If you have a discontinuous function, but it might still be integrable across the discontinuities.
The hard part about integration is that it doesn’t always come out to a known function, and it’s sometimes hard to know what that function would be. There are a variety of techniques for integrating functions when you only want the area under the entire curve. And you can always define a new function as whatever the antiderivative is of some ugly integrand, and then you have a name for it, you just don’t know much about it.
Basically what it comes down to is that you can take all the simple functions we know and differentiate them, and the answers you get are the set of functions it’s easy to invert the mapping for. All the others are hard. Because differentiation isn’t that simple a mapping between functions unless the functions themselves are relatively simple.
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