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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