# Need help with an integral involving the Dirac delta function

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?

$$\delta(g(x)) = \sum_i \frac{\delta(x-c_i)}{\vert g'(c_i) \vert},\tag{1}$$
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 $-1, 1, 2$.