Understanding Dirac delta integrals? I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$
and am told this evaluates to 8. Can anyone please step-by-step how to get there? I'm really struggling to understand how this integration works - I know that $\int \delta (x) =1$ but can't see how to apply this.
 A: In general, for "nice enough" functions (and polynomials are certainly nice enough) 
$$\int \delta(x) f(x) \,dx = f(0)$$
In some sense, the Dirac-$\delta$ is "infinite" at $x=0$, and $0$ everywhere else, so, if you pair it with other functions in an integral, it does a really good job of singling out the value of the function at $0$. (note that the Dirac delta is actually a distribution, so this "infinite at $0$" business is a little imprecise, but it'll be ok for now).
Similarly, if we were to apply simple translations, $\delta(x - c)$ would be "infinite" at $x = c$, so 
$$\int \delta(x-c) f(x) \,dx = f(c).$$
Likewise, if we also translate $f$, we get things like
$$\int \delta(x-c_1) f(x - c_2) \,dx = f(c_1 - c_2).$$
A: The idea is to evaluate the rest of the integrand at the point where the argument of $\delta$ vanishes (this only works if the argument of $\delta$ is of the form $x-c$, by the way). The argument vanishes where $x-4 = 0$, i.e., at $x=4$, so evaluate the rest of the integrand at $4$.
There is a more general formula for handling the case where the argument of $\delta$ is a product of simple linear factors with no repeated roots (google it).
