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
Through numerical methods, integration is perfectly easy. You can just go through the function and add up all the little pieces as you go along, and that gives you the integral. It is, as you’re suggesting, no harder than differentiation.
But what is hard is *analytical* integration. To get from a function to its differential, there is a specific process, df/dx = (f(x+dx)-f(x))/dx as x–>0. No matter how complicated the function f is, you can write out the formula for df/dx (if it’s a continuous, differentiable function). But integration is the reverse of that. There is no single process for it. You just have to hope that there is some function that, when differentiated, gives you f. For many functions, we know what that function is, but there’s no guarantee that an analytical function exists that can be differentiated into the function you want.
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