The value $9 \int_{0}^{9}\left[\sqrt{\frac{10x}{x+1}}\right] dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is ______.

JEE Main · 2024 · 30 Jan 2024 · Shift 1 · Definite Integrals

The value

9 \int_{0}^{9} [\frac{10 x}{x + 1}] d x

, where

[t]

denotes the greatest integer less than or equal to

t

, is ______.

Answer: 155

\frac{10 x}{x + 1} = 1, 4, 9

x = \frac{1}{9}, \frac{2}{3}, 9

respectively, so:

[\frac{10 x}{x + 1}] = 0

[0, \frac{1}{9})

= 1

[\frac{1}{9}, \frac{2}{3})

= 2

[\frac{2}{3}, 9]

I = 9 (\int_{0}^{1/9} 0 d x + \int_{1/9}^{2/3} 1 d x + \int_{2/3}^{9} 2 d x) = 9 (\frac{5}{9} + \frac{50}{3}) = 155

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