UPSC NDA I 2025 Mathematics Paper

Given,
The function is f(x)=⌊x2⌋, where ⌊x2⌋ is the greatest integer function.
We are tasked with finding:
∫‌

√3

2

√3⌊x2⌋‌dx
Decomposing the integral based on the function:
For \‌

√3

2

≤X<1,X2 lies between ‌

3

4

and 1 , so ⌊X2⌋=0. Therefore,
∫‌

√3

2

10dx=0
For 1≤x<√‌2,x2 lies between 1 and 2 , so ⌊x2⌋=1. Therefore,

√2

∫

1

1dx=√‌2−1
For √‌2≤x<√‌3,x2 lies between 2 and 3 , so ⌊x2⌋=2. Therefore,
∫√2√32dx=2(√‌3−√‌2)
Summing up all the results:
‌∫‌

√3

2

√3⌊x2⌋‌dx=0+(√‌2−1)+2(√‌3−√‌2)
‌=2(√‌3−√‌2)