Chapter 10 - The Laplace Transform and Some Elementary Applications - 10.10 Chapter Review - Additional Problems - Page 718: 10

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

Obtain: $F(s)=\int^{\infty}_0 e^{-st}f(t)dt\\ =\int^1_0 2e^{-st}dt+\int^2_1 e^{-st}(1-t) dt+\int^{\infty}_2 0e^{-st} dt\\ =[\frac{e^{-st}}{-s}]^1_0+[\frac{1-t}{-s}e^{-st}]^2_1-\frac{t}{s}[\frac{e^{-st}}{-s}]^2_1+0\\ =\frac{1}{s}(-e^{-s}+e^{-2s}+1+e^{-2s}-e^{-s}]\\ =\frac{2}{s}(e^{-2s}-e^{-s})+1$