Down the rabbit hole ......

Calculus Level 5

Let $f(x)$ be the unique, non-negative real valued function with domain $[1, \infty)$ defined by the equation

$f(x) = \displaystyle\int_{1}^{x} \dfrac{t}{f(t)} dt$ .

Now let $S = \displaystyle\int_{\sqrt{3}}^{2} f(x) dx$ . Compute $\lfloor 100 \times S \rfloor$ .