Step 1

Determine the Laplace transform of \(\displaystyle{f{{\left({t}\right)}}}={t}{\cos{{\left({2}{t}\right)}}}\)

\(L\left\{f(t)\right\}=L\left\{t \cos (2t)\right\}\)

\(\displaystyle{F}{\left({s}\right)}=-{\frac{{{d}}}{{{d}{s}}}}{\left({\frac{{{s}}}{{{s}^{{2}}+{2}^{{2}}}}}\right)}\)

\(\displaystyle=-{\frac{{{\left({s}^{{2}}+{4}\right)}-{2}{s}^{{2}}}}{{{\left({s}^{{2}}+{4}\right)}^{{2}}}}}\)

\(\displaystyle={\frac{{{s}^{{2}}-{4}}}{{{\left({s}^{{2}}+{4}\right)}^{{2}}}}}\)

step 2

Substitute 1 for s in the equation \(\displaystyle{F}{\left({s}\right)}={\frac{{{s}^{{2}}-{4}}}{{{\left({s}^{{2}}+{4}\right)}^{{2}}}}}\)

\(\displaystyle{F}{\left({1}\right)}={\frac{{{1}^{{2}}-{4}}}{{{\left({1}^{{2}}+{4}\right)}^{{2}}}}}\)

\(\displaystyle={\frac{{{1}-{4}}}{{{5}^{{2}}}}}\)

\(\displaystyle={\frac{{-{3}}}{{{25}}}}\)

Therefore \(\displaystyle{F}{\left({1}\right)}={\frac{{-{3}}}{{{25}}}}\)