Use the definition of Laplace Transform to determine the Laplace Transforms for&nbsp;the following time-domain functionf(t)=sin(3t)&nbsp;

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Use the definition of Laplace Transform to determine the Laplace Transforms for&amp;nbsp;the following time-domain functionf(t)=sin(3t)&amp;nbsp;

nick1337 · Accepted Answer

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)=∫0∞e−stf(t)dtwhere 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)=∫0∞e−stsin(3t)dtNow, we need to solve this integral using integration by parts. Let u=sin(3t), and dv=e−stdt, then du/dt=3cos(3t), and v=−1se−st.Using the integration by parts formula, we get:F(s)=[−1se−stsin(3t)]0∞+3s∫0∞e−stcos(3t)dtThe first term in the above equation evaluates to zero, since e−st 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=e−stdt, then du/dt=−3sin(3t), and v=−1se−st.Using the integration by parts formula again, we get:F(s)=9s2[−e−stcos(3t)]0∞−9s2∫0∞e−stsin(3t)dtThe 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)=9s2∫0∞e−stsin(3t)dtComparing this with the original integral, we see that F(s) appears on both sides. Solving for F(s), we get:F(s)=9s2∫0∞e−stsin(3t)dtF(s)=9s2·3s2+9Simplifying this expression, we get:F(s)=27s2(s2+9)Therefore, the Laplace Transform of f(t) = sin(3t) is given by:ℒ{sin(3t)}=F(s)=27s2(s2+9)This completes the solution.

Use the definition of Laplace Transform to determine

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