How can i solve this exercise? ### Example Problem: Laplace Transform Using Definition**Problem Statement:**Use the definition to find the Laplace Transform of the given function\[f(t) = \begin{cases} t, & 0 \le t < 1 \\ 1, & t \ge 1 \end{cases}\]**Solution Outline:**To solve this problem, you will: 1. Apply the definition of the Laplace Transform.2. Integrate the function piecewise according to its definition over the different intervals.**Step-by-Step Solution:**1. **Definition of Laplace Transform:** The Laplace Transform of a function \( f(t) \), denoted as \( \mathcal{L}\{f(t)\} \) or \( F(s) \), is given by: \[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt \]2. **Piecewise Integration:** For the given function \( f(t) \), we break the integral into two parts corresponding to the intervals of the piecewise function. \[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt = \int_{0}^{1} e^{-st} t \, dt + \int_{1}^{\infty} e^{-st} \cdot 1 \, dt \]3. **Evaluate Each Integral:** - **First Integral:** \[ \int_{0}^{1} e^{-st} t \, dt \] Use integration by parts: Let \( u = t \) and \( dv = e^{-st} dt \). Then, \( du = dt \) and \( v = -\frac{1}{s} e^{-st} \). Thus, \[ \int_{0}^{1} t e^{-st} \, dt = \left. -\frac{t}{s} e^{-st} \right|_{0}^{1} + \int_{0}^{1} \frac{1}{s} e^{-st} \, dt \] \[ = \left( -

Name: How can i solve this exercise? ### Example Problem: Laplace Transform
Uploaded: 2024-03-20T07:06:27.000Z
Channel: Bartleby
Description: How can i solve this exercise? ### Example Problem: Laplace Transform Using Definition**Problem Statement:**Use the definition to find the Laplace Transform of the given function\[f(t) = \begin{cases} t, & 0 \le t < 1 \\ 1, & t \ge 1