For any natural number $n$, the value of the integral $\mathop \smallint \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx$, is

$n\,\sqrt n \,\, - \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$n\,\sqrt n \,\, + \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$\mathop \sum \limits_{r = 1}^n \,\,\sqrt r \,\, - \,\,n\,\sqrt n $

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1

For any natural number

n

, the value of the integral

\int \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx

, is

View answer and explanation

Correct option: B

n\,\sqrt n \,\, - \,\,\sum \limits_{r = 1}^n \,\sqrt r

Option D (

n\sqrt n - \sum \limits_{r = 1}^n \,\sqrt r

) is correct because the integral is evaluated by splitting it into a sum of integrals and simplifying.

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