What is laplacetransform t^3cos(t) ?

The answer to laplacetransform t^3cos(t) is -(6(-s^4+6s^2-1))/((s^2+1)^4)

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laplacetransform t^3cos(t)

laplace transform

t^{3} cos (t)

- \frac{6 ( - s ^{4} + 6 s ^{2} - 1 )}{( s ^{2} + 1 ) ^{4}}

Solution steps

L {t^{3} cos (t)}

Use Laplace Transform table:

L {t^{k} f (t)} = (- 1)^{k} \frac{d ^{k}}{d s ^{k}} (L {f (t)})

For

t^{3} cos (t) : f (t) = cos (t), k = 3

= (- 1)^{3} \frac{d ^{3}}{d s ^{3}} (L {cos (t)})

L {cos (t)} : \frac{s}{s ^{2} + 1}

\frac{d ^{3}}{d s ^{3}} (\frac{s}{s ^{2} + 1}) = \frac{6 ( - s ^{4} + 6 s ^{2} - 1 )}{( s ^{2} + 1 ) ^{4}}

= (- 1)^{3} \frac{6 ( - s ^{4} + 6 s ^{2} - 1 )}{( s ^{2} + 1 ) ^{4}}

= - \frac{6 ( - s ^{4} + 6 s ^{2} - 1 )}{( s ^{2} + 1 ) ^{4}}

\int \frac{1}{x ^{4} + x} d x

x \to 0 lim (\frac{( 2 - x ) ^{2} - 4}{x})

x^{3} - x, 3 x

\frac{\partial}{\partial x} (x^{2} sin^{2} (y))

\frac{\partial}{\partial x} (y^{3} sin (4 x))

What is laplacetransform t^3cos(t) ?
The answer to laplacetransform t^3cos(t) is -(6(-s^4+6s^2-1))/((s^2+1)^4)