# What is the arclength of f(x)=(x-3)-ln(x/2) on x in [2,3]?

May 26, 2018

-sqrt(5)+sqrt(13)+arcsinh(3)/sqrt(2)-arcsinh(5)/sqrt(2)+log(2)+log(3)+log(-1+sqrt(5))-log(-2+sqrt(13)

#### Explanation:

• using that
$L \left(a , b\right) = {\int}_{a}^{b} \sqrt{1 + {\left(f ' \left(x\right)\right)}^{2}} \mathrm{dx}$
we have for
$f \left(x\right) = \left(x - 3\right) - \ln \left(\frac{x}{2}\right)$
$f ' \left(x\right) = 1 - \frac{2}{x} \cdot \left(\frac{1}{2}\right) = 1 - \frac{1}{x}$
and we must solve
${\int}_{2}^{3} \sqrt{1 + {\left(1 - \frac{1}{x}\right)}^{2}} \mathrm{dx}$
To computing this integral is compligated, and i will give only a result for the indefinite integral

$\frac{\sqrt{\frac{2 {x}^{2} - 2 x + 1}{x} ^ 2} \cdot x \cdot \left(- \sqrt{2} \cdot \arcsin h \left(2 x - 1\right) + 2 \sqrt{2 {x}^{2} - 2 x + 1} + 2 a c \tanh \left(\frac{x - 1}{\sqrt{2 {x}^{2} - 2 x + 1}}\right)\right)}{2 \cdot \sqrt{2 {x}^{2} - 2 x + 1}}$