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 (integer)
155
Solution
<p>$$\begin{array}{ll}
\frac{10 x}{x+1}=1 & \Rightarrow x=\frac{1}{9} \\
\frac{10 x}{x+1}=4 & \Rightarrow x=\frac{2}{3} \\
\frac{10 x}{x+1}=9 & \Rightarrow x=9
\end{array}$$</p>
<p>$$\begin{aligned}
& \mathrm{I}=9\left(\int_\limits0^{1 / 9} 0 \mathrm{dx}+\int_\limits{1 / 9}^{2 / 3} 1\mathrm{d} x+\int_\limits{2 / 3}^9 2 \mathrm{dx}\right) \\
& =155
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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