What is laplacetransform f(t)=t*sin(3t) ?

The answer to laplacetransform f(t)=t*sin(3t) is (6s)/((s^2+9)^2)

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laplacetransform f(t)=t*sin(3t)

laplace transform

f (t) = t \cdot sin (3 t)

\frac{6 s}{( s ^{2} + 9 ) ^{2}}

Solution steps

L {t sin (3 t)}

Use Laplace Transform table:

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

For

t sin (3 t) : f (t) = sin (3 t), k = 1

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

L {sin (3 t)} : \frac{3}{s ^{2} + 9}

\frac{d}{d s} (\frac{3}{s ^{2} + 9}) = - \frac{6 s}{( s ^{2} + 9 ) ^{2}}

= (- 1)^{1} (- \frac{6 s}{( s ^{2} + 9 ) ^{2}})

= \frac{6 s}{( s ^{2} + 9 ) ^{2}}

\int sinh (x) + C d x

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

f (x) = e^{3 t s i n (2 t)}

\int \frac{3}{u ^{2} + 4} d u

x \to - 1 lim (\frac{e ^{x}}{1 + x})

What is laplacetransform f(t)=t*sin(3t) ?
The answer to laplacetransform f(t)=t*sin(3t) is (6s)/((s^2+9)^2)