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An odd question (relationship between derivative and primitive)

  1. Nov 2, 2008 #1
    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.
  2. jcsd
  3. Nov 3, 2008 #2


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    Homework Helper

    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
  4. Nov 3, 2008 #3


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    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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