Answer to Question #277189 in Calculus for nitish

Question

find laplace transform : sin wt (0 < t< pi/w)

Answer 1

$sin\omega t \to \frac{\omega}{s^2+\omega^2}$

Laplace transform:

$L(f(t))=\int^{\pi/\omega}_0 e^{-st}f(t)dt=\int^{\pi/\omega}_0e^{-st}sin\omega tdt=$

$=-\frac{e^{-st}(ssin\omega t+\omega cos\omega t)}{s^2+\omega^2}|^{\pi/\omega}_0=-\frac{-\omega e^{-\pi s/\omega}-\omega}{s^2+\omega^2}=\frac{\omega e^{-\pi s/\omega}+\omega}{s^2+\omega^2}$