Laplace transformation on an exponential

Question

Using the definition of Laplace transformation (and without using a table), how to find the Laplace transformation of $$ g(t)= \begin{cases} 0,&\text{if }0\leq t\leq 4;\\ e^{3t}&\text{if }4\le t<\infty. \end{cases} $$

Answer 1

$\int_0^\infty e^{-\lambda t}f(t)dt=\int_4^\infty e^{-\lambda t} e^{3 t}dt$

$=\int_4^\infty e^{(3-\lambda) t} dt=\frac{1}{3-\lambda}\int_4^\infty e^{(3-\lambda) t} d(3-\lambda)$

$=\frac{1}{3-\lambda} \left( \left. e^{(3-\lambda)t}\right|_{t=4}^{t=\infty}\right)=\frac{1}{3-\lambda}(0-e^{(3-\lambda)4})$

for $\lambda>3$