# What is the arc length of f(x)=(x^3 + x)^5  in the interval [2,3]?

Jun 25, 2018

$\approx 2.42 \cdot {10}^{7}$

#### Explanation:

We Need the Formula

$s = {\int}_{a}^{b} \sqrt{1 + {\left(f ' \left(x\right)\right)}^{2}} \mathrm{dx}$

we have
$f ' \left(x\right) = 5 {\left({x}^{3} + x\right)}^{4} \left(3 {x}^{2} + 1\right)$
so our integral is given by

${\int}_{2}^{3} \sqrt{1 + {\left(5 {\left({x}^{3} + x\right)}^{4} \left(3 {x}^{2} + 1\right)\right)}^{2}} \mathrm{dx}$
this is $\approx 2.42 \cdot {10}^{7}$