Determine the Laplace transform of the following integrals. L\{\int_0^t e^{-2t}t^2 dt\}

Sinead Mcgee 2021-09-03 Answered

Determine the Laplace transform of the following integrals
L{0te2tt2dt}

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Expert Answer

jlo2niT
Answered 2021-09-04 Author has 96 answers

L{0te2tt2dt}
We know that , if L[f(t)]=F(s)
then L[0tf(v)dv]=F(s)s
Here we have
0te2tt2dt
so f(t)=e2tt2
Since , if L[g(t)]=G(s) then
L[eatg(t)]=G(sa)
so it we take g(t)=t2
G(s)=L[t2]=2s3
and hence
L[e2tt2]=(2s3)ss(2)
L[e2tt2]=2(s+2)3
Then
L[0te2tt2dt]=2(s+2)3s
L[0te2tt2dt]=2s(s+2)3

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