Is this correct ?
$$x(t)= \begin{cases} e^{-5t}, & 0≤t<5 \\ 0, & \text{otherwise} \end{cases}$$
\begin{align} L[x(t)] &=\int_{-\infty}^∞x(t)e^{-st}dt \\ &=\int_{0}^5e^{-5t}e^{-st} dt \\ &=\int_{0}^5e^{-(5+s)t}dt \\ &=\left[\frac{-1}{5+s}∙e^{-(5+s)t} \right]_{0}^{5} \\ &={-1\over 5+s}\left(e^{-(5+s)5}-e^{0}\right) \\ &={-1\over 5+s}\left(e^{-(5+s)5}-e^{0}\right) \\ &={-1\over 5+s}\left(e^{-(5+s)5}-1\right) \\ &={1\over 5+s}\left(1-e^{-(5+s)5}\right) \end{align}