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

The answer to inverse oflaplace 1/((s^2+s+1)) is e^{-t/2} 2/(sqrt(3))sin((sqrt(3)t)/2)

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

inverse laplace

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

e^{- \frac{t}{2}} \frac{2}{3} sin (\frac{3 t}{2})

Solution steps

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

\frac{1}{s ^{2} + s + 1} = \frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}

= L^{- 1} {\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}}}

Apply inverse transform rule:

L^{- 1} {F (s)} = f (t)

then

L^{- 1} {F (s - a)} = e^{a t} f (t)

For

\frac{1}{( s + \frac{1}{2} ) ^{2} + \frac{3}{4}} : a = - \frac{1}{2}, F (s) = \frac{1}{s ^{2} + \frac{3}{4}}

= e^{- \frac{1}{2} t} L^{- 1} {\frac{1}{s ^{2} + \frac{3}{4}}}

L^{- 1} {\frac{1}{s ^{2} + \frac{3}{4}}} : \frac{2}{3} sin (\frac{3 t}{2})

= e^{- \frac{t}{2}} \frac{2}{3} sin (\frac{3 t}{2})

What is inverse oflaplace 1/((s^2+s+1)) ?
The answer to inverse oflaplace 1/((s^2+s+1)) is e^{-t/2} 2/(sqrt(3))sin((sqrt(3)t)/2)