# Complicated delta function integral

1. Oct 29, 2012

### helpcometk

1. The problem statement, all variables and given/known data
Hi guys ,please look at the integral on the attachement.Does anyone have seen this integral before ?

2. Relevant equations
We have the following two properties :

∫δ'(x-x0)f(x) dx =-f'(x0)

δ(x^2-a^2)= {δ(x-a) +δ(x+a)}/2a

3. The attempt at a solution
Please help ,i have studied all the physics books available and im starting searching for the answer in books of history.Im desperate .I couldn't find this integral nowhere no matter how hard i would search.

~ONE GUESS OF MINE ~
so can one say :∫{f(x){δ'(x-a) +δ'(x+a)}/2a } dx and then split this integral in two :

∫{f(x)δ'(x-a)/2a + f(x)δ'(x+a)}/2a} dx and now we use the first property i gave above
to get:
{-f'(a)-f'(-a)}/2a =-f'(a)/a ~ this just a guess and in the last step i have assumed :

f'(a)=f'(-a) is this last property true ?

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2. Oct 30, 2012

### haruspex

Are you sure of that? I get {δ(x-a) +δ(x+a)}/4a

3. Oct 31, 2012

### Mute

I'd agree with the OP.

This is not true in general. For example, take f(x) = x^2. f'(x) = 2x, which does not possess the property f'(x) = f'(-x).