# What is the arc length of f(x)= xsqrt(x^3-x+2)  on x in [1,2] ?

Jul 12, 2018

$\approx 4.3712$

#### Explanation:

We are using the formual
$\setminus {\int}_{a}^{b} \sqrt{1 + f ' {\left(x\right)}^{2}} \mathrm{dx}$

by the product and the chain rule we get

$f ' \left(x\right) = \sqrt{{x}^{3} - x + 2} + x \cdot \frac{1}{2} \cdot {\left({x}^{3} - x + 2\right)}^{- \frac{1}{2}} \cdot \left(3 {x}^{2} - 1\right)$

this is

$f ' \left(x\right) = \frac{2 \cdot \left({x}^{3} - x + 2\right) + 3 {x}^{3} - x}{2 \sqrt{{x}^{3} - x + 2}}$

simplifying we get

$f ' \left(x\right) = \frac{5 {x}^{3} - 3 x + 4}{2 \sqrt{{x}^{3} - x + 2}}$
and we have to integrate

${\int}_{1}^{2} \sqrt{1 + {\left(\frac{5 {x}^{3} - 3 x + 4}{2 \sqrt{{x}^{3} - x + 2}}\right)}^{2}} \mathrm{dx}$
By a numerical method we get $4.3712$