Explanation Given data: The function, f ( t ) = ( cos ( 2 t ) + e − 4 t ) u ( t ) Formula used: Write the general expression to find the Laplace transform of f ( t ) . ℒ [ f ( t ) ] = F ( s ) = ∫ 0 − ∞ f ( t ) e − s t d t (1) Here, s is a complex variable. Write the general expression to calculate for the unit step function. u ( t ) = { 0 , t < 0 1 , t > 0 (2) Calculation: Substitute ( cos ( 2 t ) + e − 4 t ) u ( t ) for f ( t ) in equation (1). ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 − ∞ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] e − s t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 − ∞ [ cos ( 2 t ) u ( t ) ] e − s t d t + ∫ 0 − ∞ [ e − 4 t u ( t ) ] e − s t d t (3) For equation (3), apply the unit step function in equation (2). ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = ∫ 0 ∞ [ cos ( 2 t ) ] e − s t d t + ∫ 0 ∞ [ e − 4 t ] e − s t d t = ∫ 0 ∞ [ 1 2 ( e j 2 t + e − j 2 t ) ] e − s t d t + ∫ 0 ∞ e − 4 t − s t d t { ∵ cos θ = 1 2 ( e j θ + e − j θ ) } = 1 2 ∫ 0 ∞ ( e j 2 t e − s t + e − j 2 t e − s t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 ∫ 0 ∞ ( e j 2 t − s t + e − j 2 t − s t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t (4) Reduce the equation (4) as follows, ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 ∫ 0 ∞ ( e − ( s − j 2 ) t + e − ( s + j 2 ) t ) d t + ∫ 0 ∞ e − ( s + 4 ) t d t = 1 2 ∫ 0 ∞ e − ( s − j 2 ) t d t + 1 2 ∫ 0 ∞ e − ( s + j 2 ) t d t + ∫ 0 ∞ e − ( s + 4 ) t d t ℒ [ ( cos ( 2 t ) + e − 4 t ) u ( t ) ] = 1 2 [ e − ( s − j 2 ) t − ( s − j 2 ) ] 0 ∞ + 1 2 [ e − ( s + j 2 ) t − ( s + j 2 ) ] 0 ∞ + [ e − ( s + 4 ) t − ( s + 4 ) ] 0 ∞

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Chapter 15.3, Problem 3PP

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Find the Laplace transform of f(t)=(cos(2t)+e−4t)u(t).

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Chapter 15.2, Problem 2PP

Chapter 15.3, Problem 4PP

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Chapter 15 Solutions

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Find the Laplace transform of f ( t ) = ( cos ( 2 t ) + e − 4 t ) u ( t ) .

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Find the Laplace transform of f(t)=(cos(2t)+e−4t)u(t).

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Chapter 15 Solutions