Laplace transform of e^at sin bt: Illustrative example

Evaluate the Laplace transform for the function f(t)=e^{-2t}sin(4t) using the table of common Laplace transforms.

Solution

From the line 7 of the table transform:

\displaystyle \mathscr{L}\{e^{at}\sin(bt)\}=\frac{b}{(s-a)^{2}+b^{2}}

Comparing the formula above with the function we have:

a=-2,\quad b=4

Therefore:

\begin{aligned} \mathscr{L}\{f(t)\} & =\frac{4}{(s-(-2))^{2}+4^{2}} \\ & =\frac{4}{(s+2)^{2}+16} \end{aligned}

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