Animated Solution for Mathematics - Definite Integration: For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If $I={\int }_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, then the value of $9I$ is ___.

The problem asks us to evaluate the integral: The Greatest Integer Function, denoted by $[x]$, creates a staircase-like discontinuity. To solve this, we must identify the intervals where the inner function remains constant.

Let $f(x)=\sqrt{\frac{10x}{x+1}}$. We can rewrite this expression as:

As $x$ increases, the term $\frac{1}{x+1}$ decreases, meaning $1-\frac{1}{x+1}$ increases. Thus, $f(x)$ is a strictly increasing function. This ensures that the function will hit each integer value exactly once as it climbs from its start to its end.

We evaluate the boundaries of our integral. At $x=0$, $f(0)=0$. At $x=10$, $f(10)=\sqrt{\frac{100}{11}}\approx 3.015$.

Since $f(x)$ is continuous and increasing, it passes through the integers $1,2,$ and $3$. We find the transition points by solving $f(x)=k$ for $k\in \{1,2,3\}$:

1. $\sqrt{\frac{10x}{x+1}}=1\Rightarrow \frac{10x}{x+1}=1\Rightarrow 10x=x+1\Rightarrow x=\frac{1}{9}$ 2. $\sqrt{\frac{10x}{x+1}}=2\Rightarrow \frac{10x}{x+1}=4\Rightarrow 10x=4x+4\Rightarrow x=\frac{2}{3}$ 3. $\sqrt{\frac{10x}{x+1}}=3\Rightarrow \frac{10x}{x+1}=9\Rightarrow 10x=9x+9\Rightarrow x=9$

We now partition the integral into four distinct intervals based on these transition points. In each interval, $[f(x)]$ is constant:

For $x\in [0,1/9)$, $[f(x)]=0$ For $x\in [1/9,2/3)$, $[f(x)]=1$ For $x\in [2/3,9)$, $[f(x)]=2$ For $x\in [9,10]$, $[f(x)]=3$

The integral becomes a sum of areas:

Calculating the area of each rectangle: 1. $A_1=1\times \left(\frac{2}{3}-\frac{1}{9}\right)=\frac{5}{9}$ 2. $A_2=2\times \left(9-\frac{2}{3}\right)=2\times \frac{25}{3}=\frac{50}{3}$ 3. $A_3=3\times (10-9)=3$

Summing these values:

The final result requested is $9I$: