Sifting property

Integrating $\delta(t-a)$ against $f$ returns $f(a)$ -- the defining rule for the Dirac delta.

The sifting property, $\int_{-\infty}^\infty f(t)\delta(t-a)\,dt = f(a)$ , is the defining rule for the Dirac delta: multiplying by $\delta(t-a)$ and integrating "sifts out" the single value $f(a)$ from the whole function, contributing nothing from anywhere else.

Examples

$\int_{-\infty}^\infty (t^2+1)\,\delta(t-3)\,dt = 3^2+1 = 10$ , by the sifting property.