1. **Problem Statement:** Given several signals and transforms, we need to analyze and solve for Laplace transforms and regions of convergence (ROC) for the exponential signals, and understand the piecewise and sinusoidal functions. 2. **Piecewise function:** $$x(t) = \begin{cases} 1, & 0 < t < 2 \\ -1, & 2 < t < 4 \end{cases}$$ with period $T=4$. This is a periodic rectangular waveform alternating between 1 and -1 every 2 units. 3. **Sinusoidal function:** $$x(t) = \sin(\omega t)$$ This is a standard sinusoid with angular frequency $\omega$. 4. **Integral function:** $$y(t) = \int_{-\pi}^{2t} x(\tau) d\tau$$ This integral accumulates the values of $x(\tau)$ from $-\pi$ to $2t$. 5. **Discrete signal:** $$y[n] = n x[n-2]$$ This is a discrete-time signal scaled by $n$ and shifted by 2. 6. **Exponential decay with absolute value:** $$x(t) = e^{-a|t|}$$ This is an even function decaying exponentially on both sides. 7. **One-sided exponential with unit step:** $$x(t) = e^{-at} u(t)$$ Laplace transform: $$X(s) = \int_0^{\infty} e^{-at} e^{-st} dt = \int_0^{\infty} e^{-(s+a)t} dt = \frac{1}{s+a}, \quad \text{for } \mathrm{Re}(s) > -a$$ ROC: $\mathrm{Re}(s) > -a$ 8. **Negative one-sided exponential:** $$x(t) = -e^{-at} u(-t)$$ Laplace transform: $$X(s) = \int_{-\infty}^0 -e^{-at} e^{-st} dt = -\int_{-\infty}^0 e^{-(s+a)t} dt = -\left[ \frac{e^{-(s+a)t}}{-(s+a)} \right]_{-\infty}^0 = -\frac{1}{-(s+a)} = \frac{1}{s+a}$$ ROC: $\mathrm{Re}(s) < -a$ **Summary:** - For $x(t) = e^{-at} u(t)$, $X(s) = \frac{1}{s+a}$ with ROC $\mathrm{Re}(s) > -a$. - For $x(t) = -e^{-at} u(-t)$, $X(s) = \frac{1}{s+a}$ with ROC $\mathrm{Re}(s) < -a$. These results are standard Laplace transform pairs for causal and anti-causal exponentials.