What ∫_{{2}}^{{3}}f(x)dx equal to?

Correct Answer is : 2(3−2)2({3}-{2})2(3−2)

Correct Answer is : 2(3−2)2({3}-{2})2(3−2)

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UPSC NDA I 2025 Mathematics Paper

Question Numbers: 99-100

Consider the following for the two (02) items that follow:

Let f(x) = [x^{2}] where [.] is the greatest integer function.

Question : 99 of 120

Marks: +1, -0

What

∫_{\sqrt{2}}^{\sqrt{3}}f(x)dx

equal to?

Solution:

Concept:

The floor function

\lfloor x^2 \rfloor

gives the greatest integer less than or equal to

x^2

. For

x

[\sqrt{2}, \sqrt{3}]

x^2

lies between

2

and

3

, so

\lfloor x^2 \rfloor = 2

(except at the endpoint

\sqrt{3}

where it equals

3

, which does not affect the integral).

Explanation:

We need to evaluate

\int_{\sqrt{2}}^{\sqrt{3}} \lfloor x^2 \rfloor \, dx

For

\sqrt{2} \le x < \sqrt{3}

, we have

2 \le x^2 < 3

, hence

\lfloor x^2 \rfloor = 2

Thus, the integral becomes

\int_{\sqrt{2}}^{\sqrt{3}} 2 \, dx = 2 \left( \sqrt{3} - \sqrt{2} \right)

Answer:

Option B:

2(\sqrt{3}-\sqrt{2})

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