The image contains the following Laplace transforms:1. **\[\mathcal{L}\{e^{-2t} \sin(2t) + e^{3t}t^2\}\]** This expression represents the Laplace transform of a function that consists of two components: an exponentially decaying sine function and a polynomial multiplied by an exponential function.2. **\[\mathcal{L}\{(1 + e^{-t})^2\}\]** This expression is the Laplace transform of a function given by the square of the sum of 1 and an exponentially decaying term.3. **\[\mathcal{L}\{te^{2t} \cos(5t)\}\]** This expression denotes the Laplace transform of a function that includes a linear term multiplied by an exponential function and a cosine function.These transforms are commonly used in engineering and mathematics to solve differential equations and analyze systems.

We will find the required Laplace transformation.

Answered: L {e-2 sin(2t) + e³¹12} L {(1+e-¹)2} L…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The Fourier transform of a derivative of a function f'(x) is simply related to the transform of the function f(x) itself. use integration by parts and basic definition of Fourier transform to prove "The time differentiation property of Fourier transform" which states that the differentiation of a function in time domain is equivalent to the multiplication of its Fourier transform by a factor iw in frequency domain. Or Fir(t)}=in F(a) E P PD-EP
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L{cos}{s) = / -st e cos(at) dt U = cos(at) dv e^(-st)dt du -asin(at)du v = (-1/s)e^(-st) 00 (a/s)e^(-st)sin(at)dt - U = als*sin(at) dv e^(-st)dt du (a^2/s)*cos(at)dt v = (-1/s)e^(-st) 00 00 1/s a/s((-1/s)e^(-st)*sin(at)) a? 00 1/s e^(-st)cos(at)dt s2 Notice that this last integral is the same as the integral we started with. Treat this like an equation and move the integral to the left hand side: 1 e-st cos(at) dt - Therefore st cos(at) dt s/(a^2+s^2) || ||
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If det A = 10 what is det(A^25)?

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L {e-2 sin(2t) + e³¹12} L {(1+e-¹)2} L {tet cos(5t)}