Laplace transformation of piecewise function

Question

we have been given the funtion:

$$f(x)= \begin{cases} 1, & 0 1 \end{cases}$$ and asked to calculate the Laplace transformation.

I know that with the Heaviside function I can re-write it as $$f(x)=1-\mu _1(t)$$

defining $\mu _1$ to be $$\mu _1(t)= \begin{cases} 0, & t < 1 \\ 1, & t \geq 1 \end{cases}$$

but I don't know where to go after that.

Am I heading in the right direction and how do I continue?

Answer 1

$$L[f(t)]=\int_{-\infty}^{\infty}f(t)e^{-st }dt$$ $$=\int_{0}^{1}e^{-st }dt=\frac{1}{s}\big[-e^s+1\big]$$