Solving a Laplace transform of a piecewise function

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Brunoboy
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Hi everyone! :)

Have a problem here I can't solve atm. Solve the Laplace transform, when:
Printed Swedish text 'Bestäm den laplacetransformerade F(s) = L{f(t)}, när' followed by (a) f(t) defined piecewise: 1 for 0 ≤ t < 1, −1 for 1 < t < 2, 0 for t > 2.


My try:

Handwritten working of a Laplace transform of a piecewise function f(t) equal to 1 for 0 to 1, -1 for 1 to 2, and 0 for t greater than 2, with the integral from 0 to infinity of e^(-st) f(t) dt split into two integrals and evaluated, ending with e^(-st) + e^(-2st).


Would really appreciate some help!
 

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