ℒ{f(t)} =∫0^1 t e^-s t d t+∫1^∞ e^-s t d t=.(-(1)/(s) t e^-s t-(1)/(s^2) e^-s t)|0 ^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 t e^-s t d t+∫1^∞ e^-s t d t=.(-(1)/(s) t...

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

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

Key Concepts

Improper Integrals

Improper integrals extend the concept of definite integrals to unbounded intervals or functions with unbounded behavior. In many transform problems, evaluating an integral from a finite point to infinity is required. Convergence criteria, such as ensuring the exponential decay overcomes the growth of other terms, are crucial to correctly evaluate these integrals.

Laplace Transform

The Laplace transform is an integral transform that converts a function of time into a function of a complex variable. It is widely used for analyzing linear time-invariant systems and differential equations by simplifying operations such as differentiation and convolution into algebraic operations in the transform domain.

Piecewise Functions

Piecewise functions are functions defined by different expressions over different parts of their domain. In the context of integral transforms, handling piecewise functions involves breaking the integral into separate parts corresponding to each segment of the function's definition, ensuring that each segment is properly integrated over its respective domain.

Integration by Parts

Integration by parts is a fundamental technique used to evaluate integrals of products of functions. It is based on the product rule of differentiation and is particularly useful when one of the functions in the product becomes simpler upon differentiation. This method is essential for computing integrals that arise in the transformation of functions involving products of polynomials and exponential terms.

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