find laplace transform : sin wt (0 < t< pi/w)
sinωt→ωs2+ω2sin\omega t \to \frac{\omega}{s^2+\omega^2}sinωt→s2+ω2ω
Laplace transform:
L(f(t))=∫0π/ωe−stf(t)dt=∫0π/ωe−stsinωtdt=L(f(t))=\int^{\pi/\omega}_0 e^{-st}f(t)dt=\int^{\pi/\omega}_0e^{-st}sin\omega tdt=L(f(t))=∫0π/ωe−stf(t)dt=∫0π/ωe−stsinωtdt=
=−e−st(ssinωt+ωcosωt)s2+ω2∣0π/ω=−−ωe−πs/ω−ωs2+ω2=ωe−πs/ω+ωs2+ω2=-\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}=−s2+ω2e−st(ssinωt+ωcosωt)∣0π/ω=−s2+ω2−ωe−πs/ω−ω=s2+ω2ωe−πs/ω+ω
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