A recycled curve

Calculus Level 3

There exists a unique, positive-valued, non-constant, continuous and differentiable function y=f(x)y = f(x) such that

  • over any specified interval, the area between f(x)f(x) and the xx-axis is equal to the arc length of the curve, and
  • f(0)=1.f(0) = 1.

If ln2ln5f(x)dx=ab\displaystyle \int_{\ln2}^{\ln5} f(x) \, dx = \dfrac{a}{b}, where aa and bb are coprime positive integers, then find a+ba + b.


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