# The force applied against a moving object travelling on a linear path is given by F(x)=xe^x . How much work would it take to move the object over x in [2, 4] ?

Jul 20, 2016

$= {e}^{2} \left(3 {e}^{2} - 1\right)$

#### Explanation:

in one dimension:

$W = {\int}_{a}^{b} \setminus F \left(x\right) \setminus \mathrm{dx}$

$= {\int}_{2}^{4} \setminus x {e}^{x} \setminus \mathrm{dx}$

$= {\int}_{2}^{4} \setminus x \frac{d}{\mathrm{dx}} \left({e}^{x}\right) \setminus \mathrm{dx}$

and so by IBP

$= {\left[x {e}^{x}\right]}_{2}^{4} - {\int}_{2}^{4} \setminus \frac{d}{\mathrm{dx}} \left(x\right) {e}^{x} \setminus \mathrm{dx}$

$= {\left[x {e}^{x}\right]}_{2}^{4} - {\int}_{2}^{4} \setminus {e}^{x} \setminus \mathrm{dx}$

$= {\left[\left(x - 1\right) {e}^{x}\right]}_{2}^{4}$

$= 3 {e}^{4} - {e}^{2}$

$= {e}^{2} \left(3 {e}^{2} - 1\right)$