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Functional Equation and Integral

Source:

August 9, 2024
algebrafunctional equationcalculusintegration2022

Problem Statement

There is a unique continuous function ff over the positive real numbers satisfying f(4)=1f(4) = 1 and 9−(f(x))4=x2(f(x))2−2xf(x) 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) for all positive xx. Compute the value of ∫0140(f(x))3 dx\int_{0}^{140} (f(x))^3\,dx.
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