Conventions for antiderivatives notation.Am I setting myself up for a fundamental misconception if I consider antidifferentiation denoted by ∫ sign as the operation that is the inverse of the differential of a function denoted by d and that also returns an arbitrary constant C? To make things clear suppose I have a function F where d d x F ( x ) = f ( x ) for all x in some interval. Then it follows d F ( x ) = f ( x ) d x now if I have ∫ ( d F ( x ) ) = F ( x ) + C from my definition of antiderivative operation denoted by ∫ then it also means ∫ f ( x ) d x = F ( x ) + C. What do you guys think?

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Conventions for antiderivatives notation.Am I setting myself up for a fundamental misconception if I consider antidifferentiation denoted by   ∫ sign as the operation that is the inverse of the differential of a function denoted by d and that also returns an arbitrary constant C? To make things clear suppose I have a function F where       d          d      x        F  (  x  )  =  f  (  x  ) for all x in some interval. Then it follows   d  F  (  x  )  =  f  (  x  )  d  x now if I have   ∫  (  d  F  (  x  )  )  =  F  (  x  )  +  C from my definition of antiderivative operation denoted by   ∫ then it also means   ∫  f  (  x  )  d  x  =  F  (  x  )  +  C. What do you guys think?

Rafael Dillon · Accepted Answer

Step 1For a given (differentiable) function, there is another single function which is its derivative.In the other direction we are not quite so lucky: if a function is integrable there is not a single function whose derivative is the original function, but a family of functions. Each member of the family corresponds to a particular choice of constant C.Step 2So, when we say &quot;the indefinite integral of f&quot; and write &quot;   ∫  f        d  x &quot; we are referring to the entire family at once. The notation &quot;  F  (  x  )  +  C&quot; is shorthand for   {  F  (  x  )  +  C  ∣  C  ∈      R    }.If an initial condition is given, then it is possible to isolate exactly which member of the family you are looking for.

Conventions for antiderivatives notation. Am I setting myself up for a fundamental misconception if

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