How to find the Laplace transform of the following function: f(t)=te^t F(s)=int_0^(oo)(te^t e^(-st))dt What method do I use to find the integral?

grabrovi0u 2022-10-21 Answered
How to find the Laplace transform of the following function:
f ( t ) = t e t
F ( s ) = 0 ( t e t e s t ) d t
What method do I use to find the integral?
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Answers (1)

Kamden Simmons
Answered 2022-10-22 Author has 10 answers
We have:
f ( t ) = t e t
So the Laplace transform of this function is:
F ( s ) = L { f ( t ) } = 0 f ( t ) e s t d t = 0 t e t e s t d t = 0 t e ( 1 s ) t d t = lim b [ 0 b t e ( 1 s ) t d t ] = lim b ( [ t 1 1 s e ( 1 s ) t ] t = 0 b 0 b 1 1 s e ( 1 s ) t d t ) = 1 1 s lim b ( b e ( 1 s ) b 0 b e ( 1 s ) t d t ) = 1 s 1 lim b 0 b e ( 1 s ) t d t ( s > 1 ) = 1 s 1 lim b [ 1 1 s e ( 1 s ) t ] 0 b ( s > 1 ) = 1 ( s 1 ) 2 lim b [ 1 e ( 1 s ) b ] ( s > 1 ) = 1 ( s 1 ) 2 ( s > 1 )
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