Antiderivative of an even functionI'm faced with an issue in terms of antiderivatives of even and odd functions.Define f ∈ C [ − a , a ] where a &gt; 0. Let f be an even function on [-a,a] .We wish to show that ∫ − a a f ( x ) d x = 2 ∫ 0 a f ( x ) d xOn the whole, I understand how to manipulate integrals and their substitution to get this result. However, I am faced with another alternative proof, where we define F ( x ) = ∫ 0 x f ( t ) d t .for − a ≤ x ≤ a, and it is easy to show that F ( − x ) = − F ( x ), and so the proof can also proceed along these lines.However, with a different definition of F, i.e. F ( x ) = ∫ − a x f ( t ) d t ,I fail to be able to obtain the result. So my question is, why does the first definition of F work, and why does the second fail?I hope to gain more understanding of even and odd functions from this.

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Antiderivative of an even functionI&#039;m faced with an issue in terms of antiderivatives of even and odd functions.Define   f  ∈  C  [  −  a  ,  a  ] where   a  &amp;gt;  0. Let f be an even function on [-a,a] .We wish to show that      ∫          −      a        a    f  (  x  )     d  x  =  2      ∫          0              a        f  (  x  )     d  xOn the whole, I understand how to manipulate integrals and their substitution to get this result. However, I am faced with another alternative proof, where we define  F  (  x  )  =      ∫    0    x    f  (  t  )     d  t  .for   −  a  ≤  x  ≤  a, and it is easy to show that   F  (  −  x  )  =  −  F  (  x  ), and so the proof can also proceed along these lines.However, with a different definition of F, i.e.  F  (  x  )  =      ∫          −      a        x    f  (  t  )     d  t  ,I fail to be able to obtain the result. So my question is, why does the first definition of F work, and why does the second fail?I hope to gain more understanding of even and odd functions from this.

humbast2 · Accepted Answer

Explanation:Your second attempt failed because F (in second definition) no need to be either even or odd although f is an even fuction on [-a,a]. As an example consider the continuous function   f  :  [  −  1  ,  1  ]  →      R   define by   f  (  x  )  =  1 for each   x  ∈  [  −  1  ,  1  ]. Then clearly f is even.Observe that   F  (  x  )  =      ∫          −      1              x        f  (  t  )  d  t  =      ∫          −      1              x        1  d  t  =  x  +  1Therefore F is neither even nor odd.

Antiderivative of an even function I'm faced with an issue in terms of antiderivatives of even and

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