Question

The value of the integral, \int_\limits{1}^{3}\left[ x ^{2}-2 x -2\right] dx , where $[x]$ denotes the greatest integer less than or equal to $x$, is :

• A. $-\sqrt{2}-\sqrt{3}+1$
• B. $-\sqrt{2}-\sqrt{3}-1$
• C. -5
• D. -4

Answer 1

Answer: (B)

$-\sqrt{2}-\sqrt{3}-1$

Answer 2

\int_\limits{1}^{3}\left(\left[(x-1)^{2}\right]-3\right) d x =\int_\limits{1}^{2}\left[x^{2}\right]-3 \int_\limits{1}^{3} d x =\int_\limits{1}^{3} 0 d x+\int_\limits{1}^{\sqrt{2}} 1 .d x+\int_\limits{\sqrt{2}}^{\sqrt{3}} 2.d x+\int_\limits{\sqrt{3}}^{2} 3. d x-6 $=\sqrt{2}-1+2(\sqrt{3}-\sqrt{2})+3(2-\sqrt{3})-6$ $=-\sqrt{2}-\sqrt{3}-1$