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Functional equation in f(x) and f(1/x): solve for f, then integrate

This type of question, asked 2 times: 6 Apr 2023, 4 Apr 2025. All 2 questions below, oldest first.

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  1. Question 1

    6 Apr 2023

    61
    Let 5f(x)+4f(1x)=1x+3,x>05f(x) + 4f\left( \frac{1}{x} \right)=\frac{1}{x}+ 3,x > 0. Then 18∫12f(x)dx18\int_{1}^{2} f(x)dx is equal to:

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  2. Question 2

    4 Apr 2025

    143
    Let f(x)+2f(1x)=x2+5f(x) + 2f\left( \frac{1}{x} \right)=x^{2}+ 5 and 2g(x)−3g(12)=x,x>02g(x) - 3g\left( \frac{1}{2} \right)=x,x> 0. If α=∫12f(x)dx\alpha=\int_{1}^{2} f(x)dx, and β=∫12g(x)dx\beta=\int_{1}^{2} g(x)dx, then the value of 9α+β9\alpha+\beta is :

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