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$ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}=\frac{\omega }{{s}^{2}+{\omega }^{2}}$

$ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}=\frac{s}{{s}^{2}+{\omega }^{2}}$

ؖF֐

 $ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}$ $={\int }_{0}^{\infty }{e}^{-st}\mathrm{sin}\omega tdt$ @ $={{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right)}^{\text{'}}\mathrm{sin}\omega tdt$ @ $={\left[-\frac{1}{s}{e}^{-st}\mathrm{sin}\omega t\right]}_{0}^{\infty }-{{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right)\left(\mathrm{sin}\omega t\right)}^{\text{'}}dt$ @ $=\frac{1}{s}{\int }_{0}^{\infty }{e}^{-st}\left(\omega \mathrm{cos}\omega t\right)dt$ @ $=\frac{1}{s}{\int }_{0}^{\infty }{\left(-\frac{1}{s}{e}^{-st}\right)}^{\text{'}}\omega \mathrm{cos}\omega tdt$ @ $=\frac{1}{s}\left\{{\left[-\frac{1}{s}{e}^{-st}\omega \mathrm{cos}\omega t\right]}_{0}^{\infty }+{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right){\left(\omega \mathrm{cos}\omega t\right)}^{\text{'}}dt\right\}$ @ $=\frac{1}{s}\left\{{\left[-\frac{1}{s}{e}^{-st}\omega \mathrm{cos}\omega t\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\frac{1}{s}{e}^{-st}{\omega }^{2}\mathrm{sin}\omega tdt\right\}$ @ $=\frac{1}{s}\left(\frac{\omega }{s}-\frac{{\omega }^{2}}{s}{\int }_{0}^{\infty }{e}^{-st}\mathrm{sin}\omega tdt\right)$

ŁC

$ℒ\text{\hspace{0.17em}}\left\{sin\omega t\right\}={\int }_{0}^{\infty }{e}^{-st}sin\omega tdt$

ł邩C

 $ℒ\left\{\mathrm{sin}\omega t\right\}\text{\hspace{0.17em}}$ $=\frac{1}{s}\left(\frac{\omega }{s}-\frac{{\omega }^{2}}{s}\text{\hspace{0.17em}}ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}\right)$ @ $ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}\text{\hspace{0.17em}}$ $=\frac{\omega }{{s}^{2}}-\frac{{\omega }^{2}}{{s}^{2}}\text{\hspace{0.17em}}ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}$

̕ $ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}\text{\hspace{0.17em}}$ ɂĉƁC

 $ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}\left(1+\frac{{\omega }^{2}}{{s}^{2}}\right)$ $=\frac{\omega }{{s}^{2}}$ $ℒ\text{\hspace{0.17em}}\left\{\mathrm{sin}\omega t\right\}\text{\hspace{0.17em}}$ $=\frac{\omega }{{s}^{2}\left(1+\frac{{\omega }^{2}}{{s}^{2}}\right)}$ $=\frac{\omega }{{s}^{2}+{\omega }^{2}}$

ؖF]֐

 $ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}$ @ $={\int }_{0}^{\infty }{e}^{-st}\mathrm{cos}\omega tdt$ @ $={{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right)}^{\text{'}}\mathrm{cos}\omega tdt$ @ @ $={\left[-\frac{1}{s}{e}^{-st}\mathrm{cos}\omega t\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right){\left(\mathrm{cos}\omega t\right)}^{\text{'}}dt$ @ @ $={\left[-\frac{1}{s}{e}^{-st}\mathrm{cos}\omega t\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right)\left(-\omega \mathrm{sin}\omega t\right)dt$ @ @ $=\frac{1}{s}-\frac{\omega }{s}{\int }_{0}^{\infty }{e}^{-st}\mathrm{sin}\omega tdt$ @ @ $=\frac{1}{s}-\frac{\omega }{s}{{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right)}^{\text{'}}\mathrm{sin}\omega tdt$ @ @ $=\frac{1}{s}-\frac{\omega }{s}\left\{{\left[-\frac{1}{s}{e}^{-st}\mathrm{sin}\omega t\right]}_{0}^{\infty }-{\int }_{0}^{\infty }\left(-\frac{1}{s}{e}^{-st}\right){\left(\mathrm{sin}\omega t\right)}^{\text{'}}dt\right\}$ @ @ $=\frac{1}{s}-\frac{\omega }{s}\left\{{\left[-\frac{1}{s}{e}^{-st}\mathrm{sin}\omega t\right]}_{0}^{\infty }+{\int }_{0}^{\infty }\frac{1}{s}{e}^{-st}\omega \mathrm{cos}\omega tdt\right\}$ @ @ $=\frac{1}{s}-\frac{{\omega }^{2}}{{s}^{2}}{\int }_{0}^{\infty }{e}^{-st}\mathrm{cos}\omega tdt$ @

ŁC

$ℒ\text{\hspace{0.17em}}\left\{cos\omega t\right\}={\int }_{0}^{\infty }{e}^{-st}cos\omega tdt$

ł邩C

 $ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}$ $=\frac{1}{s}-\frac{{\omega }^{2}}{{s}^{2}}\text{\hspace{0.17em}}ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}$ @

̕ $ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}\text{\hspace{0.17em}}$ ɂĉƁC

 $ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}\left(1+\frac{{\omega }^{2}}{{s}^{2}}\right)$ $=\frac{1}{s}$ $ℒ\text{\hspace{0.17em}}\left\{\mathrm{cos}\omega t\right\}\text{\hspace{0.17em}}$ $=\frac{1}{s\left(1+\frac{{\omega }^{2}}{{s}^{2}}\right)}$ $=\frac{s}{{s}^{2}+{\omega }^{2}}$

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