# The force applied against a moving object travelling on a linear path is given by F(x)=3xe^x . How much work would it take to move the object over x in [2, 4] ?

Jul 1, 2016

$= 9 {e}^{4} - 3 {e}^{2}$

#### Explanation:

In the simplest sense, Work = Force X Distance

[More specifically: $W = {\int}_{C} \vec{F} \left(\vec{r}\right) \cdot d \vec{r}$]

Here, in 1 dimension, $x$, we can say that

$W = \int \mathrm{dx} q \quad F \left(x\right)$

So

$W = {\int}_{2}^{4} \mathrm{dx} q \quad 3 x {e}^{x}$

this is integration by parts ie $\int u v ' = u v - \int u ' v$

so

$u = 3 x , u ' = 3$
$v ' = {e}^{x} , v = {e}^{x}$

so we have

$W = {\left[3 x {e}^{x}\right]}_{2}^{4} - \int \mathrm{dx} q \quad 3 {e}^{x}$

$= {\left[3 x {e}^{x}\right]}_{2}^{4} - {\left[3 {e}^{x}\right]}_{2}^{4}$

$= 3 {\left[{e}^{x} \left(x - 1\right)\right]}_{2}^{4}$

$= 3 \left\{\left[{e}^{4} \left(4 - 1\right)\right] - \left[{e}^{2} \left(2 - 1\right)\right]\right\}$

$= 9 {e}^{4} - 3 {e}^{2}$