Question

2.4-4 Evaluate the following integrals: (a) $\int_{-\infty}^{\infty} g(-3\tau + a)\delta(t - \tau) d\tau$ (b) $\int_{-\infty}^{\infty} \delta(\tau)g(t - \tau) d\tau$ (c) $\int_{-\infty}^{\infty} \delta(t + 2)e^{-j\omega t} dt$ (d) $\int_{-\infty}^{1} \delta(t - 2)\sin \pi t dt$ (e) $\int_{-2}^{\infty} \delta(2t + 3)e^{-4t} dt$ (f) $\int_{-2}^{2} (t^3 + 4)\delta(1 - t) dt$ (g) $\int_{-\infty}^{\infty} g(2 - t)\delta(3 - 0.5t) dt$ (h) $\int_{-\infty}^{\infty} \cos \frac{\pi}{2}(x - 5)\delta(3x - 1) dx$ Hint: $\delta(x)$ is located at $x = 0$. For example, $\delta(1 - t)$ is located at $1 - t = 0$, that is, at $t = 1$, and so on.

          2.4-4 Evaluate the following integrals:
(a) $\int_{-\infty}^{\infty} g(-3\tau + a)\delta(t - \tau) d\tau$
(b) $\int_{-\infty}^{\infty} \delta(\tau)g(t - \tau) d\tau$
(c) $\int_{-\infty}^{\infty} \delta(t + 2)e^{-j\omega t} dt$
(d) $\int_{-\infty}^{1} \delta(t - 2)\sin \pi t dt$
(e) $\int_{-2}^{\infty} \delta(2t + 3)e^{-4t} dt$
(f) $\int_{-2}^{2} (t^3 + 4)\delta(1 - t) dt$
(g) $\int_{-\infty}^{\infty} g(2 - t)\delta(3 - 0.5t) dt$
(h) $\int_{-\infty}^{\infty} \cos \frac{\pi}{2}(x - 5)\delta(3x - 1) dx$
Hint: $\delta(x)$ is located at $x = 0$. For example, $\delta(1 - t)$ is located at $1 - t = 0$, that is, at $t = 1$, and
so on.

2.4-4 Evaluate the following integrals:
(a) ∫-∞^∞ g(-3τ + a)δ(t - τ) dτ
(b) ∫-∞^∞δ(τ)g(t - τ) dτ
(c) ∫-∞^∞δ(t + 2)e^-jω t dt
(d) ∫-∞^1δ(t - 2)sinπ t dt
(e) ∫-2^∞δ(2t + 3)e^-4t dt
(f) ∫-2^2 (t^3 + 4)δ(1 - t) dt
(g) ∫-∞^∞ g(2 - t)δ(3 - 0.5t) dt
(h) ∫-∞^∞cos(π)/(2)(x - 5)δ(3x - 1) dx
Hint: δ(x) is located at x = 0. For example, δ(1 - t) is located at 1 - t = 0, that is, at t = 1, and
so on.

Added by Consuelo O.

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