Question

Find the Laplace transform of the function f(t). f(t) = sínztif25tS8; f(t):0if t < 2 or if t > 8 Click the icon to view a short table of Laplace transforms. F (s) =

Answer 1

first find the function:

First find the function: f(t) = sin(pi t)(u(t - 2) - u(t - 8)) + 0 = sin(pi t)(u(t - 2)-u(t - 8)) f(t) = sin(pi t)u(t - 2) - sin(pi t)u(t - 8) then Laplace transform: F(s) = L{sin(pi t)u(t - 2)} - L{sin(pi t)u(t - 8)} F(s) = pi e^- 2t/s^2 + pi^2 - pi e^- 8t/s^2 + pi^2 F(s) = pi e^- 2t - pi e^- 8t/s^2 + pi^2 F(s) = pi e^- 2t(1 - pi e^- 6t)/s^2 + pi^2

$f(t)= \sin(\pi t) u(t-2)-\sin(\pi t) u(t-8)$

then Laplace transform:

$F(s) = L \left \{ \sin(\pi t) u(t-2) \right \}- L\left \{ \sin(\pi t) u(t-8) \right \}$

$F(s) = \frac{\pi e^{-2t}}{s^2+\pi^2}- \frac{\pi e^{-8t}}{s^2+\pi^2}$

$F(s) =\frac{\pi e^{-2t}-\pi e^{-8t}}{s^2+\pi ^2}$

$F(s) =\frac{\pi e^{-2t}\left (1- \pi e^{-6t} \right )}{s^2+\pi ^2}$