Question

Evaluate each of the integrals (here δ(t) is the dirac delta function) (1) ∫∞−∞e5tδ(t−2)dt= (2) ∫∞−∞cos(2t)δ(t−5)dt= (3) ∫∞0e−stcos(3t)δ(t−2)dt= (4) ∫∞0e−stt3sin(t)δ(t−3)dt\)=

Answer 1

`\delta(x)` has the property that for any continuous
`f` with support in
`\mathbb R`,


`\displaystyle\int_(-\infty)^\infty f(x)\delta(x-c)\,\mathrm dx=f(c)`

where
`c\in\mathbb R`. So we have


`\displaystyle\int_(-\infty)^\infty e^(5t)\delta(t-2)\,\mathrm dt=e^(10)`


`\displaystyle\int_(-\infty)^\infty\cos2t\,\delta(t-5)\,\mathrm dt=\cos10`

The remaining two integrals are identical to the Laplace transforms of
`\cos3t\,\delta(t-2)` and
`t^3\sin t\,\delta(t-3)`. For these, we have the property that


`\mathcal L_s\{f(t)\delta(t-c)\}=f(c)e^(-sc)`

so we get


`\displaystyle\int_0^\infty e^(-st)\cos3t\,\delta(t-2)\,\mathrm dt=\mathcal L_s\{\cos3t\,\delta(t-2)\}=\cos6\,e^(-2s)`


`\displaystyle\int_0^\infty e^(-st)t^3\sin t\,\delta(t-3)\,\mathrm dt=\mathcal L_s\{t^3\sin t\,\delta(t-3)\}=27\sin3\,e^(-3s)`