$\displaystyle L\lbrace \cos\rbrace(s)= \int_0^\infty e^{-st}\cos\!\left(at\right) dt$

$\displaystyle u =$ $dv =$

$\displaystyle du =$ $v =$

$\displaystyle =$ $\displaystyle \Bigg\vert_0^\infty - \int_0^\infty$

$u =$ $dv =$

$du =$ $v =$

$\displaystyle =$ $-\Bigg\lbrack$ $\Bigg\vert_0^\infty - \displaystyle\int_0^\infty$$\Bigg\rbrack$

$\displaystyle =$ $\displaystyle -\frac{a^2}{s^2}\int_0^\infty$

Notice that this last integral is the same as the integral we started with. Treat this like an equation and move the integral to the left hand side:

$\displaystyle\int e^{-st}\cos\!\left(at\right) dt = \frac{1}{s}$

Therefore
$\displaystyle\int e^{-st}\cos\!\left(at\right) dt =$