Question

Evaluate the following integrals:
[infinity]
∫ δ(t)f(t-r) dr
-[infinity]

Answer 1

The value of the integral
`\[ \int_(-\infty)^(\infty) \delta(t)f(t-r) \, dr \]` is equal to f(t).

Step-by-step explanation:

To evaluate the given integral, we use the property of the Dirac delta function, which states that
`\[ \int_(-\infty)^(\infty) \delta(t - a) g(t) \, dt = g(a) \]`. Applying this property to the integral
`\[ \int_(-\infty)^(\infty) \delta(t)f(t-r) \, dr \]`, we consider a = t, leading to
`\[ \int_(-\infty)^(\infty) \delta(t)f(t-r) \, dr = f(t) \].`

This result arises from the nature of the Dirac delta function, which acts as a sampling function. The integral essentially "picks out" the value of the function f(t) at the point where the argument of the delta function is zero.

In summary, the integral evaluates to the function f(t), showcasing the specialized behavior of the Dirac delta function in concentrating the contribution of the integrand at a specific point.