Given that the velocity of a particle at any time t is given by

\(\displaystyle{v}{\left({t}\right)}={2}{e}^{{{2}{t}}}\)

and it is moving in the straight line we have to find the displacement. Integrate the above Equation between the limits 2 to 4

Then

\(\displaystyle{s}={\int_{{2}}^{{4}}}{2}{e}^{{{2}{t}}}{\left.{d}{t}\right.}\)

\(\displaystyle={2}{{\left[{\frac{{{e}^{{{2}{t}}}}}{{{2}}}}\right]}_{{2}}^{{4}}}\)

\(\displaystyle={e}^{{8}}-{e}^{{4}}\)

\(\displaystyle{v}{\left({t}\right)}={2}{e}^{{{2}{t}}}\)

and it is moving in the straight line we have to find the displacement. Integrate the above Equation between the limits 2 to 4

Then

\(\displaystyle{s}={\int_{{2}}^{{4}}}{2}{e}^{{{2}{t}}}{\left.{d}{t}\right.}\)

\(\displaystyle={2}{{\left[{\frac{{{e}^{{{2}{t}}}}}{{{2}}}}\right]}_{{2}}^{{4}}}\)

\(\displaystyle={e}^{{8}}-{e}^{{4}}\)