What is inverse oflaplace 3/(s^2+2s+5) ?

The answer to inverse oflaplace 3/(s^2+2s+5) is 3/2 e^{-t}sin(2t)

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inverse oflaplace 3/(s^2+2s+5)

inverse laplace

\frac{3}{s ^{2} + 2 s + 5}

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

Solution steps

L^{- 1} {\frac{3}{s ^{2} + 2 s + 5}}

Expand

\frac{3}{s ^{2} + 2 s + 5} : 3 \cdot \frac{1}{( s + 1 ) ^{2} + 4}

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

Use the constant multiplication property of Inverse Laplace Transform:

For function

f (t)

and constant

a : L^{- 1} {a \cdot f (t)} = a \cdot L^{- 1} {f (t)}

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

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

= 3 e^{- t} \frac{1}{2} sin (2 t)

Refine

3 e^{- t} \frac{1}{2} sin (2 t) : \frac{3}{2} e^{- t} sin (2 t)

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

\frac{d}{d x} (2 x^{4} + 3 x - 3)

\int_{- \infty}^{- 1} e^{- 6 t} d t

4 sin (x), 4 cos (2 x), - \frac{π}{2}, \frac{π}{6}

\frac{d}{d x} ((x - 1) (x + 1)^{2})

x \to 0 lim (x^{l n (x)})

What is inverse oflaplace 3/(s^2+2s+5) ?
The answer to inverse oflaplace 3/(s^2+2s+5) is 3/2 e^{-t}sin(2t)