Confusion on ambiguity of antiderivatives. Would this statement, <mi mathvariant="normal">&#x22

Walker Guerrero

Walker Guerrero

Answered question

2022-05-26

Confusion on ambiguity of antiderivatives.
Would this statement, k R , be true or false? And are these interchangeable under all circumstances?
f ( x ) d x + k = f ( x ) d x
Since we could assign F(x) to either the LHS or RHS, and differentiating F(x) would both result with f(x). However if we assign I = f ( x ) d x, then I = I + k k = 0 even though we already stated "for all k". I think my misunderstanding stems from notation and what they represent. Would appreciate any clarification on this as I could not find a direct take on this flexibility of antiderivatives, especially in such context of equality.

Answer & Explanation

humanistex3

humanistex3

Beginner2022-05-27Added 9 answers

Step 1
No, your statement doesn't hold in general, I think you're misunderstanding the notation a bit. For a differentiable function F such that F = f we have that f ( x ) d x = F ( x ) + k.
Step 2
Notice that functions that differ in a constant have the same derivative, that's why we consider k when writing the antiderivative. For example take f ( x ) = x 2 and g ( x ) = x 2 + 1, then f ( x ) = 2 x = g ( x ).

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