\(\displaystyle{F}{\left({s}\right)}={\int_{{0}}^{\infty}}{e}^{{-{s}{x}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle={\int_{{0}}^{\infty}}{\cos{{\left({2}{x}\right)}}}{e}^{{{s}{x}}}{\left.{d}{x}\right.}\)

\(\displaystyle=\lim_{{{t}\to\infty}}{\int_{{0}}^{{t}}}{\cos{{\left({2}{x}\right)}}}{e}^{{{s}{x}}}{\left.{d}{x}\right.}\)

Using integration by parts we get

\(\displaystyle\int{\cos{{\left({2}{x}\right)}}}{e}^{{{s}{x}}}{\left.{d}{x}\right.}=\frac{{1}}{{{s}^{{2}}+{4}}}{e}^{{-{s}{r}}}{\left({2}{\sin{{\left({2}{x}\right)}}}-{s}{\cos{{2}}}{x}\right)}\)

Hence

\(\displaystyle{F}{\left({s}\right)}=\lim_{{{t}\to\infty}}{\left(\frac{{1}}{{{s}^{{2}}+{4}}}{e}^{{-{s}{r}}}{\left({2}{\sin{{\left({2}{x}\right)}}}-{s}{\cos{{2}}}{x}\right)}\right)}\)

\(\displaystyle=\lim_{{{t}\to\infty}}{\left(\frac{{s}}{{{s}^{{2}}+{4}}}-\frac{{1}}{{{s}^{{2}}+{4}}}{e}^{{-{s}{t}}}{\left({2}{\sin{{\left({2}{t}\right)}}}-{t}{\cos{{\left({2}{t}\right)}}}\right)}\right)}\)

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

Domain of F is \(\displaystyle{\left[{0},∞\right)}\)

Answer

\(\displaystyle{F}{\left({s}\right)}=\frac{{1}}{{s}}\)

Domain of F is \(\displaystyle{\left[{0},∞\right)}\)