The value of 9 _0^9[(10x)/(x+1)]dx, where [t] denotes the greatest integer less than or eq…

Question

The value of $9\int _{0}^{9}[\sqrt{\frac{10x}{x+1}}]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is

Answer: 155

Accepted Answer

$\begin{gathered} \frac{10x}{x+1}=1 \Rightarrow x=\frac{1}{9} \ \frac{10x}{x+1}=4 \Rightarrow x=\frac{2}{3} \ \frac{10x}{x+1}=9 \Rightarrow x=9 \end{gathered}$

$\begin{gathered} I=9((|\int _{0}^{1/9}0dx+\int _{1/9}^{2/3}1dx+\int _{2/3}^{9}2dx))| \ =155 \end{gathered}$

Solutions · 1