# An object with a mass of 4 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 5+tanx . How much work would it take to move the object over x in [(-5pi)/12, (pi)/4], where x is in meters?

May 6, 2017

As we know that work $= {\int}_{{x}_{1}}^{{x}_{2}} \vec{F} . \vec{\mathrm{dx}}$
Here $F = 4 \times 10 \times \left(5 + \tan x\right) = 40 \times \left(5 + \tan x\right)$
$\therefore W = {\int}_{-} {\left(\frac{5 \pi}{12}\right)}^{\frac{\pi}{4}} 40 \left(5 + \tan x\right) \mathrm{dx}$
$= 200 \times \left(\frac{\pi}{4} - \left(- \frac{5 \pi}{12}\right)\right) + 40 \left[\ln \sec \left(\frac{\pi}{4}\right) - \ln \sec \left(- \frac{5 \pi}{12}\right)\right]$
Now simplify it...