How do I find the Laplace transform of f(t)={(0,0<=t<=(pi)/2),(sin(t),t >= pi/2):}

Oscar Burton

Oscar Burton

Answered question

2022-10-12

How do I find the Laplace transform of
f ( t ) = { 0 0 t < π 2 sin ( t ) t π 2
Thought about using the shift theorem L { f ( t a ) H ( t a ) } ( s ) = e a s L { f ( t ) } ( s )
or will it just be the integral π 2 e s t sin ( t ) d t?

Answer & Explanation

garbhaighzf

garbhaighzf

Beginner2022-10-13Added 13 answers

Using the shifting property:
L ( f ( t ) ) ( x ) = e π s / 2 L ( cos t ) ( s ) = e π s / 2 s s 2 + 1
or directly:
L ( f ( t ) ( s ) = 0 e s t f ( t ) d t = π / 2 e s t sin t d t = sin t e s t s | π / 2 + 1 s π / 2 cos t e s t d t =
e π s / 2 s + 1 s [ cos t e s t s | π / 2 1 s π / 2 sin t e s t d t ]
( 1 + 1 s 2 ) π / 2 e s t sin t d t = e π s / 2 s π / 2 e s t sin t d t = e π s / 2 s s 2 + 1
The second way is a good practice in improper integrals and integration by parts.

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