Need help with an integral involving the Dirac delta function

Question

I'm having trouble evaluating this integral, which involves the Dirac delta function:

$$ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x} \delta \left[ (x^2-1)(x-2) \right] \mathrm{d}x $$

I think I should use the following relation:

$$\int_a^b f(t) \delta(x-c)\,\mathrm dx=\begin{cases}f(c)&\text{if }a<c<b,\\0&\text{otherwise,}\end{cases}$$

But I don't know how to apply it to this particular problem. Is there any other property of this function I should consider?

Answer 1

You have to use the property

$$\delta(g(x)) = \sum_i \frac{\delta(x-c_i)}{\vert g'(c_i) \vert},\tag{1}$$

where $c_i$ are the roots of $g$. This may look strange but the proof is not very difficult.

Also, the relation that you wrote is not correct. Once you expressed the delta as shown in (1), then you use: $$\int_a^b f(x)\delta(x-x_0) dx = \begin{cases} f(x_0) & \text{if } a<x_0<b,\\ 0 & \text{otherwise.} \end{cases}$$ Your zeros are clearly $-1, 1, 2$.