What is laplacetransform xe^{-3x}sin(5x) ?

The answer to laplacetransform xe^{-3x}sin(5x) is (5(2s+6))/((s^2+6s+34)^2)

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laplacetransform xe^{-3x}sin(5x)

laplace transform

x e^{- 3 x} sin (5 x)

\frac{5 ( 2 s + 6 )}{( s ^{2} + 6 s + 34 ) ^{2}}

Solution steps

L {e^{- 3 x} x sin (5 x)}

Use Laplace Transform table:

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

For

x e^{- 3 x} sin (5 x) : f (x) = e^{- 3 x} sin (5 x), k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{- 3 x} sin (5 x)})

L {e^{- 3 x} sin (5 x)} : \frac{5}{( s + 3 ) ^{2} + 25}

\frac{d}{d s} (\frac{5}{( s + 3 ) ^{2} + 25}) = - \frac{5 ( 2 s + 6 )}{( s ^{2} + 6 s + 34 ) ^{2}}

= (- 1)^{1} (- \frac{5 ( 2 s + 6 )}{( s ^{2} + 6 s + 34 ) ^{2}})

= \frac{5 ( 2 s + 6 )}{( s ^{2} + 6 s + 34 ) ^{2}}

x \to \infty lim (arctan (\frac{x + 1}{x + 2}))

\frac{\partial}{\partial x} (2 x^{2} + 3 y^{2})

\frac{d y}{d x} - 2 x y = 6 x

x \to 2 lim (\frac{\frac{2}{x} - 1}{x - 2})

x \to π lim (\frac{sin ( x )}{x})

What is laplacetransform xe^{-3x}sin(5x) ?
The answer to laplacetransform xe^{-3x}sin(5x) is (5(2s+6))/((s^2+6s+34)^2)