Find Laplace transform of sin(at) by definition L{sin at}=int_0^(-> +oo) e^(−st)sin at dt

jhenezhubby01ff

jhenezhubby01ff

Answered question

2022-10-05

I want to find Laplace transform of sin ( a t ) by definition
L { sin a t } = 0 + e s t sin a t d t
After taking integration by parts twice, I reached the following at last
L { sin a t } = e s t s 2 s 2 + a 2 ( 1 s sin a t + a s 2 cos a t ) | t = 0 t +
I can' t do the rest.

Answer & Explanation

Jase Powell

Jase Powell

Beginner2022-10-06Added 11 answers

You are forgetting to evaluate e s t as t :
L { sin a t } = s 2 s 2 + a 2 [ e s t ( 1 s sin a t + a s 2 cos a t ) ] t = 0 t =
What happens to the expression in square brackets when t ?
beninar6u

beninar6u

Beginner2022-10-07Added 1 answers

L { sin a t } = 0 + e s t sin a t d t
I = 0 + e s t sin a t = 1 2 i 0 + e s t ( e i a t e i a t ) d t = 1 2 i 0 + ( e t ( i a + s ) e t ( i a + s ) d t
2 i I = | e t ( i a + s ) ( i a s ) | 0 + | e t ( i a + s ) ( i a + s ) | 0
2 i I = 1 ( s i a ) 1 ( s + i a )
I = a s 2 + a 2

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