# Let F(s) be the Laplace tranform of f(x)=t cos (2t) find the value of F(1)

Let F(s) be the Laplace tranform of $f\left(x\right)=t\mathrm{cos}\left(2t\right)$
find the value of F(1)
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Step 1
Determine the Laplace transform of $f\left(t\right)=t\mathrm{cos}\left(2t\right)$
$L\left\{f\left(t\right)\right\}=L\left\{t\mathrm{cos}\left(2t\right)\right\}$
$F\left(s\right)=-\frac{d}{ds}\left(\frac{s}{{s}^{2}+{2}^{2}}\right)$
$=-\frac{\left({s}^{2}+4\right)-2{s}^{2}}{{\left({s}^{2}+4\right)}^{2}}$
$=\frac{{s}^{2}-4}{{\left({s}^{2}+4\right)}^{2}}$
step 2
Substitute 1 for s in the equation $F\left(s\right)=\frac{{s}^{2}-4}{{\left({s}^{2}+4\right)}^{2}}$
$F\left(1\right)=\frac{{1}^{2}-4}{{\left({1}^{2}+4\right)}^{2}}$
$=\frac{1-4}{{5}^{2}}$
$=\frac{-3}{25}$
Therefore $F\left(1\right)=\frac{-3}{25}$