NDA Mathematics Question (2025) — Solution

Step-by-step solution

The given integral is _ 2 ^ 2 [x^2] \, dx. As x varies from 2 to 2, x^2 varies from 2 to 4. The greatest integer function [x^2] changes its value when x^2 is an integer. The integer values of x^2 in this interval are 2, 3, and 4, which correspond to x = 2 , 3 , and 2. We split the integral at x = 3 : For x ( 2 , 3 ), 2 For x ( 3 , 2), 3 The integral can be written as: _ 2 ^ 2 [x^2] \, dx = _ 2 ^ 3 2 \, dx + _ 3 ^ 2 3 \, dx Evaluating the integrals: _ 2 ^ 3 2 \, dx = 2[x]_ 2 ^ 3 = 2( 3 - 2 ) = 2 3 - 2 2 _ 3 ^ 2 3 \, dx = 3[x]_ 3 ^ 2 = 3(2 - 3 ) = 6 - 3 3 Adding the two values: _ 2 ^ 2 [x^2] \, dx = (2 3 - 2 2 ) + (6 - 3 3 ) _ 2 ^ 2 [x^2] \, dx = 6 - 3 - 2 2 Answer: 6 - 3 - 2 2