How to solve this differential equation involving an integral

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I am working on a problem involving finding a function for which the area under the curve is equal to the arc length. I therefore come up with $\int_a^b(k\sqrt{1+[f'(x)]²} -f(x))dx =0$. Frankly, I have no idea where to even start to solve this. I have read that it may involve hyperbolic functions, but I'm not really sure.