# Math Help - Laplace Transformation

1. ## Laplace Transform

I'm a little a bit confused about the following exercise because of the two segments of the function. How can we find the Laplace transformation of this function

$f(t) = \begin {cases} t , 0\le t < 4 \\ 5 , t\ge 4\end {cases}$

2. Just use the definition of the Laplace transform:
$F(s)=\int_0^{\infty}e^{-st}f(t)dt$
$=\int_0^{4}e^{-st}\cdot{t}\,dt+\int_{4}^{\infty}e^{-st}\cdot{5}\,dt$
You should be able to figure out those integrals.

--Kevin C.

3. The simplest way is to solve the integral...

$\int_{0}^{\infty} f(t)\cdot e^{- s\cdot t}\cdot dt = \int_{0}^{4} t\cdot e^{- s\cdot t}\cdot dt + \int_{4}^{\infty} 5\cdot e^{- s\cdot t}\cdot dt =$

$= - \frac{1}{s}\cdot |t\cdot e^{- s\cdot t}|_{0}^{4} - \frac{1}{s^{2}}\cdot |e^{- s\cdot t}|_{0}^{4} - \frac{5}{s}\cdot |e^{- s\cdot t}|_{4}^{\infty} = \frac{1- (1-s)\cdot e^{-4\cdot s}}{s^{2}}$

Kind regards

$\chi$ $\sigma$