The value 9 _0^9 [ 10 x x+1 ] d x, where [t] denotes the greatest… - Vidyadip

MCQ

The value $9 \int_0^9\left[\sqrt{\frac{10 x}{x+1}}\right] d x$, where $[t]$ denotes the greatest integer less than or equal to $t$, is___________.

✓
$155$
B
$166$
C
$444$
D
$421$

✓

Answer

Correct option: A.

$155$

a
$\frac{10 x}{x+1}=1$ $\Rightarrow x=\frac{1}{9}$

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

$\frac{10 x}{x+1}=9$ $\Rightarrow x=9$

$\mathrm{I}=9\left(\int_0^{1 / 9} 0 \mathrm{dx}+\int_{1 / 9}^{2 / 3} 1 . \mathrm{dx}+\int_{2 / 3}^9 2 \mathrm{dx}\right)$

$=155$

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