Question

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

Answer 1

`-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(a,b)=int_a^bsqrt(1+(f'(x))^2)dx`
  we have for
  `f(x)=(x-3)-ln(x/2)`
  `f'(x)=1-2/x*(1/2)=1-1/x`
  and we must solve
  `int_2^3sqrt(1+(1-1/x)^2)dx`
  To computing this integral is compligated, and i will give only a result for the indefinite integral

`(sqrt((2x^2-2x+1)/x^2)*x*(-sqrt(2)*arcsinh(2x-1)+2sqrt(2x^2-2x+1)+2actanh((x-1)/sqrt(2x^2-2x+1))))/(2*sqrt(2x^2-2x+1))`