Definite Integral Involving Greatest Integer Function of Square Root Expression

To evaluate the integral $I = \int_{0}^{10} \left[ \sqrt{\frac{10x}{x + 1}} \right] dx$, we analyze the values of $f(x) = \sqrt{\frac{10x}{x + 1}}$ on the interval $[0, 10]$.

The function $f(x)$ is strictly increasing on $[0, 10]$ with $f(0) = 0$ and $f(10) = \sqrt{\frac{100}{11}} \approx 3.015$. The greatest integer function $[f(x)]$ changes its integer value at points where $f(x) = 1, 2, 3$:

1. $f(x) = 1 \implies x = \frac{1}{9}$
2. $f(x) = 2 \implies x = \frac{2}{3}$
3. $f(x) = 3 \implies x = 9$

Splitting the integral across these critical points: $I = \int_{0}^{1/9} 0 \, dx + \int_{1/9}^{2/3} 1 \, dx + \int_{2/3}^{9} 2 \, dx + \int_{9}^{10} 3 \, dx$

Evaluating each piece yields: $I = 0 + \left( \frac{2}{3} - \frac{1}{9} \right) \cdot 1 + \left( 9 - \frac{2}{3} \right) \cdot 2 + (10 - 9) \cdot 3 = \frac{5}{9} + \frac{50}{3} + 3 = \frac{182}{9}$

Thus, the value of $9I$ is $182$.