Complicated delta function integral

[helpcometk]

Homework Statement

Hi guys ,please look at the integral on the attachement.Does anyone have seen this integral before ?

Homework 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

The Attempt at a Solution

Please help ,i have studied all the physics books available and I am 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 ?

This was just a guess ,and i need a definite answer so please don't reply if you are not sure about the answer.

 

[haruspex]

δ(x^2-a^2)= {δ(x-a) +δ(x+a)}/2a

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

 

[Mute]

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

I'd agree with the OP.

~ this just a guess and in the last step i have assumed :

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

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