Find inverse laplace transform of the following:
$$L^{-1}\{{\frac{e^{-s}}{s(s+1)}}\}$$
I have try to decompose the fraction into
$$\frac{A}{s}+\frac{B}{s+1}$$
now $e^{-s}$ as numerator is new for me, may I know how to proceed?
2026-03-28 19:29:06.1774726146
Inverse laplace e^-s/(s(s+1))
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Leave the $e^{-s}$ alone. We see $$\frac{e^{-s}}{s(s+1)} = e^{-s} \left(\frac{1}{s(s+1)} \right) = e^{-s}\left(\frac{1}{s} - \frac{1}{s+1} \right) = \frac{e^{-s}}{s} - \frac{e^{-s}}{s+1}.$$ Now you can take the inverse transform of the two terms separately.