Problem 02
By using the linearity property, show that
$\mathcal{L}(\cosh at) = \dfrac{s}{s^2 - a^2}$
Solution 02
$f(t) = \cosh at$
$\displaystyle \mathcal{L}\left\{ f(t) \right\} = \int_0^\infty e^{st} f(t) \, dt$
$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \cosh at \, dt$
But
$\cosh at = \dfrac{e^{at} + e^{-at}}{2}$
Thus,
$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \left( \dfrac{e^{at} + e^{-at}}{2} \right) \, dt$
