The value of _ ,0 ^ , 2 [ x^2 ] ,dx , where [.] is the greatest i… - Vidyadip

MCQ

The value of $\int_{\,0}^{\,\sqrt 2 } {[{x^2}]\,dx} ,$ where $[.]$ is the greatest integer function

A
$2 - \sqrt 2 $
B
$2 + \sqrt 2 $
✓
$\sqrt 2 - 1$
D
$\sqrt 2 - 2$

✓

Answer

Correct option: C.

$\sqrt 2 - 1$

c
(c) $I = \int_0^{\sqrt 2 } {[{x^2}]\,\,dx} $

$ = \int_{\,0}^{\,1} {[{x^2}]\,dx + } \int_{\,1}^{\,\sqrt 2 } {[{x^2}]\,\,dx} $

$ = \int_{\,0}^{\,1} {\,0\,dx + } \int_{\,1}^{\,\sqrt 2 } {\,dx} $

$ = [x]_1^{\sqrt 2 } = \sqrt 2 - 1$.

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