Antiderivative of an odd function Is the antiderivative of an odd function even? The answer given

Mayra Berry

Mayra Berry

Answered question

2022-06-20

Antiderivative of an odd function
Is the antiderivative of an odd function even?
The answer given by the book is yes.
However, I found a counterexample defined in R { 0 }:
f ( x ) = { ln | x | + 1 x < 0 ln | x | x > 0
Its derivative is 1 x , which is an odd function.
Question: is my counterexample right?

Answer & Explanation

iceniessyoy

iceniessyoy

Beginner2022-06-21Added 27 answers

Step 1
I think that f should be defined on an interval I which contains 0 and is symmetric to 0. If F is an antiderivative of f on I, then there is a constant c such that F ( x ) = 0 x f ( t ) d t + c ..
Step 2
If you now calculate F ( x ) with the substitution s = t you will get F ( x ) = F ( x ) ..

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