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

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

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