Evaluate the integrals int-infty^infty e^cos t delta(t-pi) d...

Evaluate the integrals int_{-infty}^{infty} e^{cos t} delta(t-pi) d t int_{1}^{infty} e^{-2 x} delta(x) d x int_{-infty}^{infty} e^{-t} delta(t+3) d t int_{-infty}^{infty} delta(2 t-4)left(2 t^{2}+t-8 ight) d t int_{-infty}^{infty} cos (9 t) delta(t-2) d t

Transcript

00:01 In this problem, we want to evaluate five integrals that all require the use of the delta dirac function.

00:07 So let's first go through some properties of the delta dirac distribution.

00:16 So we will call it a function, but mathematically speaking, it's mostly a distribution.

00:21 But i digress.

00:24 The delta dirac has the following properties that makes it very useful in an integral.

00:30 So the delta dirac of t minus a is equal to zero everywhere except for when t is equal to a.

00:49 And this property is particularly useful when you want to integrate, because if you wanted to evaluate the integral, let's say some function f of t times our delta dirac of t minus a dt, because our dirac collapses everywhere to zero outside of when t is equal to a, this makes our integral collapse to zero everywhere except for when t is equal to a.

01:25 So the resultant integral corresponds to our function f evaluated at a.

01:34 Because we're going to need it, we have another property.

01:37 The delta dirac of some constant a times t is equal to one over the absolute value of a times the delta dirac of t.

01:57 There's this yelling's constant, and it's a property that we're going to utilize at some point, so i'm writing it down, but it's not the most obvious of properties.

02:13 These first two are by far what's most used in a delta dirac calculation.

02:21 So let's go through our first integral.

02:23 We want to evaluate the integral of e to the cos of t times delta t minus pi for t ranging from minus infinity to plus infinity...

Question

Evaluate the integrals int_{-infty}^{infty} e^{cos t} delta(t-pi) d t int_{1}^{infty} e^{-2 x} delta(x) d x int_{-infty}^{infty} e^{-t} delta(t+3) d t int_{-infty}^{infty} delta(2 t-4)left(2 t^{2}+t-8 ight) d t int_{-infty}^{infty} cos (9 t) delta(t-2) d t

Please give Ace some feedback