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}