What is laplacetransform 2((e^{2t}+e^{-2t})/2)sin(2t) ?

The answer to laplacetransform 2((e^{2t}+e^{-2t})/2)sin(2t) is 2/((s-2)^2+4)+2/((s+2)^2+4)

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laplacetransform 2((e^{2t}+e^{-2t})/2)sin(2t)

laplace transform

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

\frac{2}{( s - 2 ) ^{2} + 4} + \frac{2}{( s + 2 ) ^{2} + 4}

Solution steps

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

Expand

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

= L {e^{2 t} sin (2 t) + e^{- 2 t} sin (2 t)}

Use the linearity property of Laplace Transform:

For functions

f (t), g (t)

and constants

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {e^{2 t} sin (2 t)} + L {e^{- 2 t} sin (2 t)}

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

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

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

\frac{\partial}{\partial x} (z^{2} x^{3})

f (x) = x^{2} + 3

\int 2 arctan (x) d x

f (x) = x^{5} + 4, g (x) = x + 4

f (x) = - \frac{1}{2 1 - x}, at x = - 8

What is laplacetransform 2((e^{2t}+e^{-2t})/2)sin(2t) ?
The answer to laplacetransform 2((e^{2t}+e^{-2t})/2)sin(2t) is 2/((s-2)^2+4)+2/((s+2)^2+4)