If f ( x ) is even, then what can we say about: <msubsup> &#x222B;<!-- ∫ --

Poftethef9t

Poftethef9t

Answered question

2022-06-20

If f ( x ) is even, then what can we say about:
2 2 f ( x ) d x
If f ( x ) is odd, then what can we say about
2 2 f ( x ) d x
Are they both zero? For the first one if its even wouldn't this be the same as
a a f ( x ) d x = 0
Now if its odd f ( x ) = f ( x ). Would FTOC make this zero as well?

Answer & Explanation

iceniessyoy

iceniessyoy

Beginner2022-06-21Added 27 answers

If f ( x ) is even then f ( x ) = f ( x ). So
2 2 f ( x ) d x = 2 0 f ( x ) d x + 0 2 f ( x ) d x = 0 2 f ( x ) d x + 0 2 f ( x ) d x
But then f ( x ) = f ( x ) f ( x ) = f ( x ) so that simplifies to 2 0 2 f.
Similarly, if f is odd - that is: f ( x ) = f ( x ) we get
2 2 f ( x ) d x = 2 0 f ( x ) d x + 0 2 f ( x ) d x = 0 2 f ( x ) d x + 0 2 f ( x ) d x = 0
Emanuel Keith

Emanuel Keith

Beginner2022-06-22Added 8 answers

Not exactly:
{ a a f ( x ) d x = 2 0 a f ( x ) d x if  f  is even, a a f ( x ) d x = 0 if  f  is odd.
To see it, make the substitution t = x, d x = d t:
a 0 f ( x ) d x = a 0 f ( t ) d t = 0 a f ( t ) d t = { 0 a f ( t ) d t ( f even ) , 0 a f ( t ) d t ( f odd ) ,
then use Chasles relation.

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