Finding the Laplace Transformation of $|t-1|$ (absolute value)

Question

I am unsure how to find the Laplace Transformation to this. I tried breaking the limit from $0$-infinity up, but it did not help.

Answer 1

When $t \in [0,1]$, $|t-1| = 1-t$; otherwise, it is $t-1$. The LT is then

$$\hat{f}(s) = \int_0^1 dt \: (1-t) e^{-s t} + \int_1^{\infty} dt \: (t-1) e^{-s t}$$

Use the fact that

$$\int dt \: t\,e^{-s t} = -\frac{e^{-s t} (s \,t+1)}{s^2}$$

so that

$$\hat{f}(s) =\frac{1-2 e^{-s}}{s} + \frac{2(s+1) e^{-s}-1}{s^2} = \frac{1}{s} + \frac{2 e^{-s}-1}{s^2}$$