Evaluate the following integrals involving the impulse function: [latex]\int_{0}^{10}\left(t^{2}+4 t-2\right) \delta(t-2) d t[/latex] [latex]\int_{-\infty}^{\infty}\left(\delta(t-1) e^{-t} \cos t+\delta(t+1) e^{-t} \sin t\right) d t[/latex]

For the first integral, we apply the sifting property in Eq. (7.32). [latex]\int_{a}^{b} f(t) \delta\left(t-t_{0}\right) d t=f\left(t_{0}\right)[/latex] (7.32) [latex]\int_{0}^{10}\left(t^{2}+4 t-2\right) \delta(t-2) d t=\left.\left(t^{2}+4 t-2\right)\right|_{t=2}=4+8-2=10[/latex] Similarly, for the second integral, [latex]\int_{-\infty}^{\infty}\left(\delta(t-1) e^{-t} \cos t+\delta(t+1) e^{-t} \sin t\right) d t[/latex] [latex]=\left.e^{-t} \cos t\right|_{t=1}+\left.e^{-t} \sin t\right|_{t=-1}[/latex] [latex]=e^{-1} \cos 1+e^{1} \sin (-1)=0.1988-2.2873=-2.0885[/latex]

Question 7.9: Evaluate the following integrals involving the impulse funct...

Fundamentals of Electric Circuits

Evaluate the following integrals involving the impulse function:

$\int_{0}^{10}\left(t^{2}+4 t-2\right) \delta(t-2) d t$
$\int_{-\infty}^{\infty}\left(\delta(t-1) e^{-t} \cos t+\delta(t+1) e^{-t} \sin t\right) d t$

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For the first integral, we apply the sifting property in Eq. (7.32).

$\int_{a}^{b} f(t) \delta\left(t-t_{0}\right) d t=f\left(t_{0}\right)$ (7.32)

\int_{0}^{10}\left(t^{2}+4 t-2\right) \delta(t-2) d t=\left.\left(t^{2}+4 t-2\right)\right|_{t=2}=4+8-2=10

Similarly, for the second integral,

$\int_{-\infty}^{\infty}\left(\delta(t-1) e^{-t} \cos t+\delta(t+1) e^{-t} \sin t\right) d t$
$=\left.e^{-t} \cos t\right|_{t=1}+\left.e^{-t} \sin t\right|_{t=-1}$
$=e^{-1} \cos 1+e^{1} \sin (-1)=0.1988-2.2873=-2.0885$