• question_answer The value of the integral$\int_{-10}^{1}{\frac{\left| \frac{2[x]}{3x-[x]} \right|}{\frac{2[x]}{3x-[x]}}},$ where [ ] represents GIF A)  0                                             B)  10                    C)  -10                                         D)  $10-\frac{2}{3}$

Idea $\because$ [x] = n for $n\le x<n+1$ such as [2.45] = 2, $[-2.1]=-3,[0.32]=0$ $\int_{a}^{b}{f(x)dx}=\int_{a}^{c}{f(x)}dx+\int_{c}^{d}{f(x)dx}+\int_{a}^{b}{f(x)}dx$ According to given question, we know that f(x) is not define at $x=0,-\frac{1}{3}$ $\therefore$$\int_{-10}^{1}{\frac{\frac{2[x]}{3x-[x]}}{\frac{2[x]}{3x-[x]}}}$ $=\int_{-10}^{-1/3}{dx+}\int_{-1/3}^{0}{(-1)}dx+\int_{0}^{1}{(0)}dx$                 $=[x]_{-10}^{-1/3}+[-x]_{-1/3}^{0}+0$ $=-\frac{1}{3}+10+0-\frac{1}{3}$            $=10-\frac{2}{3}$ TEST Edge Integration of real function based questions are asked. To solve these types of questions, students are advised to understand the concept of integration and also acquainted yourself with the real functions.
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