Q232 / 270 · NDA 2018 (Phase 1) · Mathematics

What is $\int_0^{\sqrt{2}}\left[x^2\right] d x$ equal to (where [.] is the greatest integer function)?

  • Correct answer
    A
    $\sqrt{2}-1$
  • B
    $1-\sqrt{2}$
  • C
    $2(\sqrt{2}-1)$
  • D
    $\sqrt{3}-1$

Solution

$\begin{aligned} & \int_0^{\sqrt{2}}\left[x^2\right] \mathrm{dx} \\ & =\int_0^1\left[x^2\right] d x+\int_1^{\sqrt{2}}\left[x^2\right] \mathrm{dx}\end{aligned}$ $=0+\int_1^{\sqrt{2}} 1 . \mathrm{dx} \quad(\because[]$ is greatest integer function $)$ $=(\mathrm{x})_1^{\sqrt{2}}=\sqrt{2}-1$

Hence, the correct answer is option: 'A'.

Relevant Questions