Encontrar a transformada de Laplace de $f(t) = \cos t$.

$\mathcal{L}\{f(t)\} = \displaystyle\int_0^{+\infty} f(t)e^{-st}\ dt = \displaystyle\int_0^{+\infty} e^{-st}\cos t\ dt = \left.\left(e^{-st}\sin t\right)\right|_0^{+\infty} + s\displaystyle\int_0^{+\infty} e^{-st}\sin t\ dt =$

$= \left.\left(e^{-st}\sin t\right)\right|_0^{+\infty} - \left.\left(se^{-st}\cos t\right)\right|_0^{+\infty} - s^2\underset{\mathcal{L}\{\cos t\}}{\underbrace{\displaystyle\int_0^{+\infty} e^{-st}\cos t\ dt}}$

$\mathcal{L}\{\cos t\} = \dfrac{\left.\left(e^{-st}\sin t\right)\right|_0^{+\infty} - \left.\left(se^{-st}\cos t\right)\right|_0^{+\infty}}{1 + s^2}$, que converge para $s > 0$.

$\fbox{$\mathcal{L}\{\cos t\} = \dfrac{s}{1 + s^2},\ s > 0$}$