What is laplacetransform x*sin(x) ?

The answer to laplacetransform x*sin(x) is (2s)/((s^2+1)^2)

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laplacetransform x*sin(x)

laplace transform

x \cdot sin (x)

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

Solution steps

L {x sin (x)}

Use Laplace Transform table:

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

For

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

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

L {sin (x)} : \frac{1}{s ^{2} + 1}

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

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

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

y^{^{''}} - 12 y^{^{'}} + 36 y = 0, y (0) = 1, y (1) = 0

\int \frac{x ^{2}}{( x - 3 ) ^{3}} d x

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = 0

f (x) = x^{3} - 7 x^{2}, at x = 0

\int \frac{x ^{6}}{x ^{14} - 7} d x

What is laplacetransform x*sin(x) ?
The answer to laplacetransform x*sin(x) is (2s)/((s^2+1)^2)