Let's do some calculus! (20)

Calculus Level 4

\[\lim_{n \to \infty} \left(\sum_{k=1}^n a^{k-1} \int_{(k-1)a}^{ka} {\dfrac{f(x)}{f(x) + f \left( \left(2k-1\right)a-x \right)}} \ dx \right) = \dfrac{7}{5} \]

Given that function \(f(x) > 0\), \(\forall ~ x \in \mathbb{R}\) is bounded and satisfies the condition above. If \(a < 1\), find \(a\).

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\[\dfrac{9}{14}\] \[\dfrac{1}{7}\] \[\dfrac{1}{19}\] \[\dfrac{2}{7}\] \[\dfrac{14}{9}\] \[\dfrac{9}{19}\] \[\dfrac{14}{19}\] \[\dfrac{7}{19}\]

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