What is inverse oflaplace 1/(s^2+2*3*pi*s+pi^2) ?

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

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inverse oflaplace 1/(s^2+23pi*s+pi^2)

inverse laplace

\frac{1}{s ^{2} + 2 \cdot 3 \cdot π \cdot s + π ^{2}}

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

Solution steps

L^{- 1} {\frac{1}{s ^{2} + 2 \cdot 3 π s + π ^{2}}}

\frac{1}{s ^{2} + 23 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)

\int \frac{x}{7 x + 1} d x

\frac{\partial}{\partial x} (250 x + 370 y - 100000)

\int \frac{1}{1 + e ^{- x}} d x

x \to + 2 - lim (\frac{x}{x ^{2} + 4})

\frac{d}{d x} ((- x + \frac{π}{2}) tan (x))

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