What is laplacetransform f(x)=xe^{2x}sin(6x) ?

The answer to laplacetransform f(x)=xe^{2x}sin(6x) is (6(2s-4))/((s^2-4s+40)^2)

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laplacetransform f(x)=xe^{2x}sin(6x)

laplace transform

f (x) = x e^{2 x} sin (6 x)

\frac{6 ( 2 s - 4 )}{( s ^{2} - 4 s + 40 ) ^{2}}

Solution steps

L {e^{2 x} x sin (6 x)}

Use Laplace Transform table:

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

For

x e^{2 x} sin (6 x) : f (x) = e^{2 x} sin (6 x), k = 1

= (- 1)^{1} \frac{d}{d s} (L {e^{2 x} sin (6 x)})

L {e^{2 x} sin (6 x)} : \frac{6}{( s - 2 ) ^{2} + 36}

\frac{d}{d s} (\frac{6}{( s - 2 ) ^{2} + 36}) = - \frac{6 ( 2 s - 4 )}{( s ^{2} - 4 s + 40 ) ^{2}}

= (- 1)^{1} (- \frac{6 ( 2 s - 4 )}{( s ^{2} - 4 s + 40 ) ^{2}})

= \frac{6 ( 2 s - 4 )}{( s ^{2} - 4 s + 40 ) ^{2}}

\frac{d}{d x} (ln (x^{2} - 1))

x \to 4 lim (x^{2} - 16 x - 4)

\int \frac{x}{5 + x ^{2}} d x

\frac{d y}{d x}, y = x^{2} sin (8 x)

\int \frac{1}{x 4 - x ^{2}} d x

What is laplacetransform f(x)=xe^{2x}sin(6x) ?
The answer to laplacetransform f(x)=xe^{2x}sin(6x) is (6(2s-4))/((s^2-4s+40)^2)