# What is the arc length of f(x)=-xln(1/x)-xlnx on x in [3,5]?

Mar 13, 2017

$2$

#### Explanation:

Use the rule $\log \left({a}^{b}\right) = b \log \left(a\right)$ to simplify the function:

$f \left(x\right) = - x \ln \left({x}^{-} 1\right) - x \ln \left(x\right)$

Bringing the $- 1$ out:

$f \left(x\right) = x \ln \left(x\right) - x \ln \left(x\right)$

$f \left(x\right) = 0$

This is the straight line $y = 0$. Thus its arc length on $x \in \left[3 , 5\right]$ is just the line segment with length $2$.

Using $f \left(x\right) = 0$ we can apply the arc length formula for $f$ on $x \in \left[a , b\right]$ for the same result:

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

$L = {\int}_{2}^{4} \sqrt{1 + 0} \mathrm{dx}$

$L = {\int}_{2}^{4} \mathrm{dx}$

L=x]_2^4

$L = 2$