A recycled curve

Calculus Level 3

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

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

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

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