JEE MAIN - Mathematics (2024 - 30th January Morning Shift - No. 23)
The value of $$9 \int_\limits0^9\left[\sqrt{\frac{10 x}{x+1}}\right] \mathrm{d} x$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is
Answer
155
Explanation
$$\begin{array}{ll} \frac{10 x}{x+1}=1 & \Rightarrow x=\frac{1}{9} \\ \frac{10 x}{x+1}=4 & \Rightarrow x=\frac{2}{3} \\ \frac{10 x}{x+1}=9 & \Rightarrow x=9 \end{array}$$
$$\begin{aligned} & \mathrm{I}=9\left(\int_\limits0^{1 / 9} 0 \mathrm{dx}+\int_\limits{1 / 9}^{2 / 3} 1\mathrm{d} x+\int_\limits{2 / 3}^9 2 \mathrm{dx}\right) \\ & =155 \end{aligned}$$
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