What is laplacetransform te^{-4t} ?

The answer to laplacetransform te^{-4t} is 1/((s+4)^2)

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laplacetransform te^{-4t}

laplace transform

t e^{- 4 t}

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

Solution steps

L {e^{- 4 t} t}

Use Laplace Transform table:

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

For

t e^{- 4 t} : f (t) = e^{- 4 t}, k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{- 4 t}})

L {e^{- 4 t}} : \frac{1}{s + 4}

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

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

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

What is laplacetransform te^{-4t} ?
The answer to laplacetransform te^{-4t} is 1/((s+4)^2)