delta(t^2 - 1) = 0.5 for t = 1 or -1
i think the main idea is that the integration of any delta from -∞ to ∞ must be 1 , so the weight added is to asure that area is 1 and not 2 if we do the integration on ur fromula.
delta function represents impulse of action at the origin of system. For example when we push a key it is like an impulse of force at that instant(say t=0, key pushing time). When this delta is multiplied with some other function it makes that function an impulse function at zeros of that function i. e given function shifts origin of impulse. For example key pressed at t=0, t=1,t=2 can be represented by