To evaluate the integral

$\int_0^1 \left( \sqrt{-1 + \sqrt{\frac{4}{x}} - 3} \right) \, dx$

we need to carefully examine the expression inside the square roots, as this integral might not be well-defined over the entire interval $[0, 1]$ due to the presence of square roots and potential undefined values.

Step 1: Analyze the Inner Expression

Inside the square root, we have: $-1 + \sqrt{\frac{4}{x}} - 3.$

Simplify the expression: $\sqrt{\frac{4}{x}} - 4.$

For the square root to be real and non-negative, $\sqrt{\frac{4}{x}} - 4 \geq 0$, which gives constraints on $x$ that need to be checked to see if