What is laplacetransform x^3e^{-3x} ?

The answer to laplacetransform x^3e^{-3x} is 6/((s+3)^4)

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laplacetransform x^3e^{-3x}

laplace transform

x^{3} e^{- 3 x}

\frac{6}{( s + 3 ) ^{4}}

Solution steps

L {e^{- 3 x} x^{3}}

Use Laplace Transform table:

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

For

x^{3} e^{- 3 x} : f (x) = e^{- 3 x}, k = 3

= (- 1)^{3} \frac{d ^{3}}{d s ^{3}} (L {e^{- 3 x}})

L {e^{- 3 x}} : \frac{1}{s + 3}

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

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

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

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

f (x) = (- x^{2} + 4)^{2}

y = \frac{2}{x - 2} (4.1)

x \to 0 + lim ((2 x)^{5 x})

\int \frac{1}{y ^{2}} d y

What is laplacetransform x^3e^{-3x} ?
The answer to laplacetransform x^3e^{-3x} is 6/((s+3)^4)