Integral of odd or even functions?

[xdrgnh]

Is the integral of an even function an odd function and vice versa? I know the derivative is.

 

[slider142]

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:
$$\int_0^{-x} f(t) dt = -\int_0^x f(t) dt$$
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 ?

 

[xdrgnh]

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.

 

[Jorriss]

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.

 

[HallsofIvy]

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!