Q77 / 120 · VITEEE 2014 · Mathematics
The value of $\int_{0}^{\sqrt{2}}\left[x^{2}\right] d x$, where $[.]$ is the greatest integer function, is
- A$2-\sqrt{2}$
- B$2+\sqrt{2}$
- Correct answerC$\sqrt{2}-1$
- D$\sqrt{2}-2$
Solution
$\int_{0}^{\sqrt{2}}\left[x^{2}\right] d x=\int_{0}^{1}\left[x^{2}\right] d x+\int_{1}^{\sqrt{2}}\left[x^{2}\right] d x$
$=\int_{0}^{1} 0 \mathrm{dx}+\int_{1}^{\sqrt{2}} 1 \mathrm{dx}$
$=[\mathrm{x}]_{1}^{\sqrt{2}}=\sqrt{2}-1$
Hence, the correct answer is option: 'C'.