What is inverse oflaplace s/((s+1)^2) ?

The answer to inverse oflaplace s/((s+1)^2) is e^{-t}-e^{-t}t

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inverse oflaplace s/((s+1)^2)

inverse laplace

\frac{s}{( s + 1 ) ^{2}}

e^{- t} - e^{- t} t

Solution steps

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

Take the partial fraction of

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

= L^{- 1} {\frac{1}{s + 1} - \frac{1}{( s + 1 ) ^{2}}}

Use the linearity property of Inverse Laplace Transform:

For functions

f (s), g (s)

and constants

a, b : L^{- 1} {a \cdot f (s) + b \cdot g (s)} = a \cdot L^{- 1} {f (s)} + b \cdot L^{- 1} {g (s)}

= L^{- 1} {\frac{1}{s + 1}} - L^{- 1} {\frac{1}{( s + 1 ) ^{2}}}

L^{- 1} {\frac{1}{s + 1}} : e^{- t}

L^{- 1} {\frac{1}{( s + 1 ) ^{2}}} : e^{- t} t

= e^{- t} - e^{- t} t

\int \frac{4 x ^{5}}{1 - 12 x ^{6}} d x

12 x^{3} - 18 x^{2}

\frac{d y}{d x} + 2 y = k

f (x) = e^{x^{2}}, at x = \frac{1}{2}

What is inverse oflaplace s/((s+1)^2) ?
The answer to inverse oflaplace s/((s+1)^2) is e^{-t}-e^{-t}t