Dirac delta translation property: product of functions in the integrand

Question

Given $\int f(x) \delta(x-a)\,\mathrm dx = f(a)$, does it follow that $\int f(x) \delta(x-a) g(x)\,\mathrm dx = f(a)g(a)$?

https://en.wikipedia.org/wiki/Dirac_delta_function#Translation

Answer 1

Define $f(x)g(x)=h(x)$, then, we have $$\int f(x)\delta(x-a) g(x)dx=\int h(x)\delta (x-a)dx\\ =h(a)=f(a)g(a)$$