Problem 02
Find the Laplace transform of $f(t) = e^{at}$.
Solution 02
$\displaystyle \mathcal{L} \left\{f(t)\right\} = \int_0^\infty e^{-st} f(t) \, dt$
$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-st} e^{at} \, dt$
$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-st + at} \, dt$
$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-(s - a)t} \, dt$
$\displaystyle \mathcal{L} (e^{at}) = -\dfrac{1}{s - a} \int_0^\infty e^{-(s - a)t} \, [ \, -(s - a) \, dt \, ]$