An odd question (relationship between derivative and primitive)

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The discussion centers on the relationship between derivatives and antiderivatives, emphasizing that while derivatives can be directly calculated from their definitions, antiderivatives represent an "inverse problem" that lacks a general explicit formula. The antiderivative is defined as the function whose derivative is the given function, making it inherently more complex. The conversation highlights the distinction between direct problems, such as calculating a function's value, and inverse problems, like finding roots of equations or antiderivatives, which are generally more challenging.

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Is there an explicit formula for finding the antiderivative of a function? I was thinking that perhaps it would be the inverse function of the derivative, but I don't know what that would be off the top of my head.
 
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There are (with some restrictions), but they are not useful

one is the definite integral
F=∫f dx
another is differentiation can be effected by some opperator A
F=(A^-1)f
 
This is an example of the difference between "direct" and "inverse" methods.

We have a specific definition of the "derivative" of a function and so, can, theoretically, find the derivative of any given (differentiable) function directly from that definition. The anti-derivative of the function f, on the other hand, is only defined as "the function that has f as its derivative". The first is a "direct problem" since we are given a direct definition and formula, the second is an "inverse problem" since the anti-derivative is only defined as an inverse of the derivative. "Inverse problems" are typically much harder than "direct problems".

The same situation occurs in elementary algebra. If I define f(x)= x7- 3x5+ x4- 4x+ 5 and as "what is f(1)", that's easy: just set x= 1 and calculate: 1- 3+ 1- 4+ 5= 0, because I gave you the formula. If, instead, I ask you to solve the equation f(x)= 0, that is much harder- there is no general formula for solving such an equation. Having just calculated f(1)= 0 tells you that x= 1 is one solution, just as having learned that the derivative of x3 is 3x2 tells you that x3 is one anti-derivative of 3x2, but you still don't know if there are other solutions.
 

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