Find the Laplace transforms of the functions given in problem f(t)=sin pi t text{ if } 2leq tleq3 , f(t)=0 text{ if } t<2 text{ or if } t>3

lwfrgin 2020-11-23 Answered
Find the Laplace transforms of the functions given in problem
f(t)=sinπt if 2t3,
f(t)=0 if t<2 or if t>3
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Expert Answer

ensojadasH
Answered 2020-11-24 Author has 100 answers
Step 1
Given function is
\(\begin{cases}\sin \pi t & 2\leq t\leq3\0 & 23\end{cases}\)
The Laplace transform of function f(t) is defined by
L[f(t)]=0estf(t)dt
=02estf(t)dt+23estf(t)dt+3estf(t)dt
=23estf(t)dt{f(t)=0,2<t or t>3}
=23estsin(πt)dt
Step 2
Now, we firstly evaluate the indefinite integral by parts.
I=estsin(πt)dt=sin(πt)estdt((ddt)sin(πt)estdt)dt
I=estssin(πt)πcos(πt)(ests)dt
I=estssin(πt)+πsestcos(πt)dt
Again , we integrate by parts.
I=estssin(πt)+πs[cos(πt)estdt(ddtcosπtestdt)dt]
I=estssin(πt)+πs[estscos(πt)(πsin(πt))(ests)dt]
I=estssin(πt)+πs[estscos(πt)πsestsin(πt)dt]
I=estssin(πt)πests2cos(πt)π2s2I
I+π2s2I=estssin(πt)πests2cos(πt)
I=estπ2+s2[s(sin(πt))+πcos(πt)]
Step 3
Now, we evaluate the Laplace transform of given function.
L[f(t)]=[estπ2+s2[s(sin(πt))+πcos(πt)]]23
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