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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Definite Integration

NDA 2018 · Mathematics · Definite Integration

What is ∫_0^ 2 [x^2] d x equal to (where [.] is the greatest...

What is ∫02[x2]dx\int_0^{\sqrt{2}}\left[x^2\right] d x equal to (where [.] is the greatest integer function)?

Options

  1. A2−1\sqrt{2}-1
  2. B1−21-\sqrt{2}
  3. C2(2−1)2(\sqrt{2}-1)
  4. D3−1\sqrt{3}-1

Correct answer

A. 2−1\sqrt{2}-1

Step-by-step solution

∫02[x2]dx and =∫01[x2]dx+∫12[x2]dx\begin{aligned}& \int_0^{\sqrt{2}} \left[x^2\right]\mathrm{dx} \\ \text{ and } =\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[]isgreatestintegerfunctionis greatest integer function)=(\mathrm{x})_1^{\sqrt{2}}=\sqrt{2}-1$
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