What is inverse oflaplace 1/((s^2+6pis+pi^2)) ?

The answer to inverse oflaplace 1/((s^2+6pis+pi^2)) is e^{-3pit} 1/(2sqrt(2)pi)sinh(2sqrt(2)pit)

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inverse oflaplace 1/((s^2+6pis+pi^2))

inverse laplace

\frac{1}{( s ^{2} + 6 π s + π ^{2} )}

e^{- 3 π t} \frac{1}{2 2 π} sinh (22 π t)

Solution steps

L^{- 1} {\frac{1}{( s ^{2} + 6 π s + π ^{2} )}}

\frac{1}{s ^{2} + 6 π s + π ^{2}} = \frac{1}{( s + 3 π ) ^{2} - 8 π ^{2}}

= L^{- 1} {\frac{1}{( s + 3 π ) ^{2} - 8 π ^{2}}}

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 + 3 π ) ^{2} - 8 π ^{2}} : a = - 3 π, F (s) = \frac{1}{s ^{2} - 8 π ^{2}}

= e^{- 3 π t} L^{- 1} {\frac{1}{s ^{2} - 8 π ^{2}}}

L^{- 1} {\frac{1}{s ^{2} - 8 π ^{2}}} : \frac{1}{2 2 π} sinh (22 π t)

= e^{- 3 π t} \frac{1}{2 2 π} sinh (22 π t)

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What is inverse oflaplace 1/((s^2+6pis+pi^2)) ?
The answer to inverse oflaplace 1/((s^2+6pis+pi^2)) is e^{-3pit} 1/(2sqrt(2)pi)sinh(2sqrt(2)pit)