A) \[5+\sqrt{2}+\sqrt{3}\]
B) \[-5+\sqrt{2}-\sqrt{3}\]
C) \[5-\sqrt{2}-\sqrt{3}\]
D) \[-4+\sqrt{3}-\sqrt{2}\]
Correct Answer: C
Solution :
\[I=\int\limits_{0}^{2}{[{{x}^{2}}]}dx\]The function [x2] varies as follows between x = (0,2) \[[{{x}^{2}}]=\left\{ \begin{align} & 0\,\text{if}\,0\le {{x}^{2}}<1,\text{or}\,0\le x<1 \\ & 1\,\text{if}\,1\le {{x}^{2}}<2\,\text{or1}\le x<\sqrt{2} \\ & 2\,\text{if}\,2\le {{x}^{2}}<3\,\text{or}\,\sqrt{2}\le x<\sqrt{3} \\ & 3\,\text{if}\,3\le {{x}^{2}}<4\text{or}\,\sqrt{3}\le x<2 \\ \end{align} \right.\] \[\Rightarrow I=\int\limits_{0}^{1}{0.dx}+\int\limits_{1}^{\sqrt{2}}{1.}dx+\int\limits_{\sqrt{2}}^{\sqrt{3}}{2.dx}+\int\limits_{\sqrt{3}}^{2}{.3dx}\] \[=0+(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})\] \[=(\sqrt{2}-1+2\sqrt{3}-2\sqrt{2}+6-3\sqrt{3}=5-\sqrt{2}-\sqrt{3}\]
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