Solving simple dirac delta function

[jwang34]

1. Homework Statement
$$\int$$ x[delta(x)-delta(x/3+4)] dx

Homework Equations

so I'm supposed to use this principle:
$$\int$$ f(x)delta(x-xo)dx=f(xo)

The Attempt at a Solution

So it seems simple but I just want to make sure that I'm applying the above principle correctly.

I separate the terms so the problem becomes:
$$\int$$ x[delta(x)]dx - $$\int$$ x[delta(x/3+4)]dx

Now the first term will go to zero because xo=0. The second term, I'm a little unsure. Should I transform the inside of the delta function so that delta(x/3+4) becomes delta(x+12)? Or should I transform the f(x) from x to x/3, then use that principle? Any suggestions are greatly appreciated.

 

[Galileo]

Use a substitution and a test function to find out what delta(ax) is in terms of delta(x).
It's a very useful relation to know in general and makes this problem easy once you got it.

Hint: Split the cases a>0 and a<0.

 

[jwang34]

ok...here's what I found

So delta(ax)=[delta(x)]/|a|. Then could I do the following:

Given delta(x/3+4)=[delta(x+12)]/|3|. Basically, I factored out the 1/3, and so a=1/3.

 

[Galileo]

Right, but then you get 3*delta(x), not delta(x)/3.