The force applied against a moving object travelling on a linear path is given by F(x)=2x+xe^x . How much work would it take to move the object over x in [1,2 ] ?

Dec 31, 2017

Answer:

$W = 3 + {e}^{2}$ Joule

Explanation:

$F \left(x\right) = 2 x + x {e}^{x}$

As we know,

$\mathrm{dW} = \vec{F} \cdot \vec{\mathrm{dx}}$

Integrating both sides

$W = {\int}_{1}^{2} \left(2 x + x {e}^{x}\right) \mathrm{dx}$

$W = {\left[{x}^{2}\right]}_{1}^{2} + {\int}_{1}^{2} x {e}^{x} \mathrm{dx}$

Applying by-parts by ILATE rule,

$W = {\left[{x}^{2}\right]}_{1}^{2} + {\left[x {e}^{x}\right]}_{1}^{2} - {\int}_{1}^{2} {e}^{x} \mathrm{dx}$

$W = {\left[{x}^{2}\right]}_{1}^{2} + {\left[x {e}^{x}\right]}_{1}^{2} - {\left[{e}^{x}\right]}_{1}^{2}$

$W = 4 - 1 + 2 {e}^{2} - e - {e}^{2} + e$ Joule

$W = 3 + {e}^{2}$ Joule