For the functions $f(t) = e^{t}$ and $g(t) = e^{-3t}$, defined on $0\leq t < \infty$, compute $f \ast g$ in two different ways:

1. By directly evaluating the integral in the definition of $f \ast g$.

$\displaystyle (f \ast g)(t) = \int_0^t$ $dw \ = \$

2. By computing ${\mathcal L}^{-1} \left\lbrace F(s)G(s) \right\rbrace$ where $F(s) = {\mathcal L} \left\lbrace f(t) \right\rbrace$ and $G(s) = {\mathcal L} \left\lbrace g(t) \right\rbrace$.

$\displaystyle (f \ast g)(t) = {\mathcal L}^{-1} \left\lbrace F(s)G(s) \right\rbrace = {\mathcal L}^{-1} \big\lbrace$ $\big\rbrace$

$\displaystyle \phantom{(f \ast g)(t)} =$ help (formulas)