Use the definition of Laplace Transform to determine

Answered question

2022-04-21

Use the definition of Laplace Transform to determine the Laplace Transforms for 
the following time-domain function

f(t)=sin(3t)

 

Answer & Explanation

nick1337

nick1337

Expert2023-04-29Added 777 answers

To find the Laplace Transform of the time-domain function f(t) = sin(3t), we will use the definition of Laplace Transform, which is given by:
F(s)=0estf(t)dt
where F(s) is the Laplace Transform of f(t), and s is a complex variable.
Substituting f(t) = sin(3t) in the above formula, we get:
F(s)=0estsin(3t)dt
Now, we need to solve this integral using integration by parts. Let u=sin(3t), and dv=estdt, then du/dt=3cos(3t), and v=1sest.
Using the integration by parts formula, we get:
F(s)=[1sestsin(3t)]0+3s0estcos(3t)dt
The first term in the above equation evaluates to zero, since est and sin(3t) both approach zero as t approaches infinity and zero.
Now, we need to evaluate the second term using integration by parts again. Let u=cos(3t), and dv=estdt, then du/dt=3sin(3t), and v=1sest.
Using the integration by parts formula again, we get:
F(s)=9s2[estcos(3t)]09s20estsin(3t)dt
The first term in the above equation evaluates to zero, since e^{-st} and cos(3t) both approach zero as t approaches infinity and zero.
Substituting the value of F(s) obtained in the second term above, we get:
F(s)=9s20estsin(3t)dt
Comparing this with the original integral, we see that F(s) appears on both sides. Solving for F(s), we get:
F(s)=9s20estsin(3t)dt
F(s)=9s2·3s2+9
Simplifying this expression, we get:
F(s)=27s2(s2+9)
Therefore, the Laplace Transform of f(t) = sin(3t) is given by:
{sin(3t)}=F(s)=27s2(s2+9)
This completes the solution.

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