The given f (t) can be written in the following form by a standard property.
f (t) = sint + [sin2t – sint] u(t – π) + [sin3t – sin2t] u(t – 2π)
Now L [f (t)] = L (sint) + L {[sin2t – sint] u(t – π)} + L{[sin3t – sin2t] u(t – 2π)} ...(1)
Consider L [sin2t – sint] u(t – π)
After this you can solve on your own.
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