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$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

For each positive integer $n$, defined $f_n(x)=$ minimum $\left(\frac{x^n}{n !}, \frac{(1-x)^n}{n !}\right)$, for $0 \leq x \leq 1 .$ Let $I_n=\int \limits_0^1 f_n(x) d x, n \geq 1$. Then, $\sum \limits_{n=1}^{\infty} I_n$ is equal to

$\int_{}^{} {\frac{{\cos 2x}}{{{{(\cos x + \sin x)}^2}}}\;dx = } $

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}x&0&8\\4&1&3\\2&0&x\end{array}\,} \right| = 0$ are equal to

The probability distribution of a random variable $X$ is:

X	0	1	2	3	4
P(X)	0.1	k	2k	k	0.1

where $k$ is some unknown constant.
The probability that the random variable $X$ takes the value 2 is:

If $g(f(x)) = |\sin x|$ and $f(g(x)) = {(\sin \sqrt x )^2}$, then

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is

$\Delta = \left| {\,\begin{array}{*{20}{c}}a&{a + b}&{a + b + c}\\{3a}&{4a + 3b}&{5a + 4b + 3c}\\{6a}&{9a + 6b}&{11a + 9b + 6c}\end{array}\,} \right|$where $a = i,b = \omega ,c = {\omega ^2}$, then $\Delta $is equal to

Let $y$ be the function which passes through $(1, 2)$ having slope $(2x + 1)$. The area bounded between the curve and $x -$ axis is

A and B are two events. $P ( A )=\frac{1}{2}, P ( B )=\frac{1}{3}$ and $P ( A \cap B )=\frac{1}{4}$ then $P \left( A ^{\prime} / B \right)=$ _________.

If $F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$ and $[F(x)]^2=F(k x)$, then the value of $k$ is :