An odd question (relationship between derivative and primitive)

[Reedeegi]

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.

 

[lurflurf]

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

 

[HallsofIvy]

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.