Animated Solution for Mathematics - Definite Integration: The value $9{\int }_0^9[\sqrt{\frac{10x}{x+1}}]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is _____

The Greatest Integer Function (GIF), denoted by $[...]$, acts like a staircase. Instead of attempting a complex integration, we identify the points where the function $f(x)=\sqrt{\frac{10x}{x+1}}$ jumps between integer values.

As $x$ increases from $0$ to $9$, the function $f(x)$ climbs monotonically from $f(0)=0$ to $f(9)=3$. Because the function is continuous and increasing, it must pass through the integer values $1$ and $2$.

To determine the intervals where the GIF remains constant, we solve for the values of $x$ where the function hits these integers.

For the first jump, we set the function equal to $1$:

Squaring both sides yields:

For the second jump, we set the function equal to $2$:

Squaring both sides yields:

We now partition the interval $[0,9]$ based on these jump points to evaluate the integral $I=9{\int }_0^9\left[\sqrt{\frac{10x}{x+1}}\right]dx$:

1. On the interval $[0,1/9]$, the function value is between $0$ and $1$, so the GIF is $0$. 2. On the interval $[1/9,2/3]$, the function value is between $1$ and $2$, so the GIF is $1$. 3. On the interval $[2/3,9]$, the function value is between $2$ and $3$, so the GIF is $2$.

The integral becomes the sum of three distinct areas:

Calculating the lengths of these intervals: - The first part is $0$. - The second part is $1\times (\frac{2}{3}-\frac{1}{9})=\frac{6-1}{9}=\frac{5}{9}$. - The third part is $2\times (9-\frac{2}{3})=2\times (\frac{27-2}{3})=2\times \frac{25}{3}=\frac{50}{3}$.

Summing these values and multiplying by the external factor of $9$:

The final result is $155$.