What is laplacetransform x*sin(Ax) ?

The answer to laplacetransform x*sin(Ax) is (2As)/((s^2+A^2)^2)

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laplacetransform x*sin(Ax)

laplace transform

x \cdot sin (A x)

\frac{2 A s}{( s ^{2} + A ^{2} ) ^{2}}

Solution steps

L {x sin (A x)}

Use Laplace Transform table:

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

For

x sin (A x) : f (x) = sin (A x), k = 1

= (- 1)^{1} \frac{d}{d s} (L {sin (A x)})

L {sin (A x)} : \frac{A}{s ^{2} + A ^{2}}

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

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

= \frac{2 A s}{( s ^{2} + A ^{2} ) ^{2}}

\frac{d}{d x} (- 2 x^{3} + 10 - 3 x^{2} + 12 x)

\frac{1}{( s ^{4} - 9 )}

\int (2 x sin (x^{2})) d x

\int sin (x) x d x

x y^{2} y^{^{'}} = 2 - y^{3}

What is laplacetransform x*sin(Ax) ?
The answer to laplacetransform x*sin(Ax) is (2As)/((s^2+A^2)^2)