ℒ{f(t)} =∫0^1-e^-s t d t+∫1^∞ e^-s t d t=.(1)/(s) e^-s t|0 ^1-.(1)/(s) e^-s t|1 ^∞ =(1)

step 1 Hello. So here we let F of X be equal to the integral from 0 to infinity of E to negative xT tim

ℒ{f(t)} =∫0^1-e^-s t d t+∫1^∞ e^-s t d t=.(1)/(s) e^-s...

$$\begin{aligned} \mathscr{L}\{f(t)\} &=\int_{0}^{1}-e^{-s t} d t+\int_{1}^{\infty} e^{-s t} d t=\left.\frac{1}{s} e^{-s t}\right|_{0} ^{1}-\left.\frac{1}{s} e^{-s t}\right|_{1} ^{\infty} \\ &=\frac{1}{s} e^{-s}-\frac{1}{s}-\left(0-\frac{1}{s} e^{-s}\right)=\frac{2}{s} e^{-s}-\frac{1}{s}, \quad s>0 \end{aligned}$$

$$\begin{aligned}
\mathscr{L}\{f(t)\} &=\int_{0}^{1}-e^{-s t} d t+\int_{1}^{\infty} e^{-s t} d t=\left.\frac{1}{s} e^{-s t}\right|_{0} ^{1}-\left.\frac{1}{s} e^{-s t}\right|_{1} ^{\infty} \\
&=\frac{1}{s} e^{-s}-\frac{1}{s}-\left(0-\frac{1}{s} e^{-s}\right)=\frac{2}{s} e^{-s}-\frac{1}{s}, \quad s>0
\end{aligned}$$

Key Concepts

Linearity Property

The linearity property of the Laplace transform states that the transform of a linear combination of functions is equal to the same linear combination of their transforms. This property is crucial when dealing with piecewise functions or functions that can be decomposed into simpler parts, as it permits separate integration and subsequent summation of the results.

Improper Integration

Improper integration involves evaluating integrals with one or more infinite limits or discontinuities in the integrand. When the Laplace transform is computed, integrals extending to infinity must be carefully handled to ensure convergence, typically under conditions such as s > 0.

Laplace Transform

The Laplace transform is an integral transform that converts a time-domain function into a complex frequency-domain representation. It is defined as an integral from zero to infinity of the product of the function and an exponential decay term, allowing differential equations to be transformed into algebraic equations and facilitating the analysis of system dynamics.

Piecewise Defined Functions

In many applications, functions are defined differently over separate intervals. When computing the Laplace transform of a piecewise function, the integral must be segmented according to the different definitions of the function. This allows each segment to be handled according to its own expression before combining the results.

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