Integral of odd or even functions?

In summary, the conversation discusses the relationship between the integral of an even function and an odd function, and whether it is necessary to start the integral from 0 or if any antiderivative can be used. The possibility of proving this relationship using known facts about integrals is brought up, but one participant mentions not having time to prove it. The other participant reminds them of the importance of verifying information, regardless of their expertise.
  • #1
xdrgnh
417
0
Is the integral of an even function an odd function and vice versa? I know the derivative is.
 
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  • #2
That is an interesting hypothesis. Try to prove it. Suppose f(t) is an arbitrary even function that is integrable on the interval [0, x]. That means all we know about the function f is that its definite integral exists, and that f(t) = f(-t). Can we use only these two facts to show that the integral over that interval F(x) is an odd function? (F(-x) = -F(x) ?) In other words, we need to prove the following equation is true:
[tex]\int_0^{-x} f(t) dt = -\int_0^x f(t) dt[/tex]
One technique would be to try to use what we know about integrals to simplify the expression on the left side to see if it is equal to the expression on the right side. Is it necessary to start our integral from 0, or can we use any antiderivative of f ?
 
  • #3
I'm a physics major who needs to calculate a Fourier series for a driven oscillator I got no time to prove it. But thanks anyway.
 
  • #4
xdrgnh said:
I'm a physics major who needs to calculate a Fourier series for a driven oscillator I got no time to prove it. But thanks anyway.
Thats your loss then. It would do you good.
 
  • #5
So if someone told you this was true (or false) what reason would you have to believe them? Just because you are a physicist, that does not mean you should not check what you are told!
 

1. What is an odd function?

An odd function is a mathematical function that satisfies the property f(-x) = -f(x). This means that when the input of the function is multiplied by -1, the output is equal to the negative of the original output. Graphically, an odd function has symmetry about the origin.

2. What is an even function?

An even function is a mathematical function that satisfies the property f(-x) = f(x). This means that when the input of the function is multiplied by -1, the output remains the same. Graphically, an even function has symmetry about the y-axis.

3. What is the integral of an odd function over a symmetric interval?

The integral of an odd function over a symmetric interval is equal to 0. This is because the positive and negative areas of the function cancel each other out, resulting in a net area of 0. This property is known as the odd function integral property.

4. What is the integral of an even function over a symmetric interval?

The integral of an even function over a symmetric interval is equal to twice the integral of the function over one half of the interval. This is because the positive and negative areas of the function are equal, so they can be combined to find the total area. This property is known as the even function integral property.

5. Can an odd or even function have a non-zero integral over a non-symmetric interval?

Yes, an odd or even function can have a non-zero integral over a non-symmetric interval. This is because the symmetry property only applies to symmetric intervals. When the interval is not symmetric, the positive and negative areas of the function do not cancel each other out.

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