# what is the laplace transformation of the function in the picture f(x)=xsin x

what is the laplace transformation of the function in the picture $f\left(x\right)=x\mathrm{sin}x$
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Step 1 Given that $f\left(x\right)=x\mathrm{sin}x$ find out the Laplace transform of $f\left(x\right)$ Step 2 Here we use the property $L\left[{t}^{n}f\left(t\right)\right]=\left(-1{\right)}^{n}\frac{{d}^{n}}{d{s}^{n}}\left[f\left(s\right)\right]$
Let $f\left(t\right)=\mathrm{sin}t$
$L\left[f\left(t\right)\right]=\frac{1}{{s}^{2}+1}=\overline{f}\left(s\right)$
$L\left[tf\left(t\right)\right]=\left(-1{\right)}^{1}\frac{d}{ds}\left[\frac{1}{{s}^{2}+1}\right]$
$=-1\frac{d}{ds}\left({s}^{2}+1{\right)}^{-1}$
$=-1\left(-1\right)\left({s}^{2}+1{\right)}^{-2}\left(2s\right)$
$=\frac{2s}{\left({s}^{2}+1{\right)}^{2}}$