_ ,0 ^ , /2 x - [ x] ,dx is equal to - Vidyadip

MCQ

$\int_{\,0}^{\,\pi /2} {\{ x - [\sin x]\} \,dx} $ is equal to

✓
$\frac{{{\pi ^2}}}{8}$
B
$\frac{{{\pi ^2}}}{8} - 1$
C
$\frac{{{\pi ^2}}}{8} - 2$
D
None of these

✓

Answer

Correct option: A.

$\frac{{{\pi ^2}}}{8}$

a
(a) $\int_0^{\pi /2} {\{ x - [\sin x]\} \,dx = \int_0^{\pi /2} {xdx - \int_{\,0}^{\,\pi /2} {[\sin x]\,\,dx} } } $

$ = \left( {\frac{{{x^2}}}{2}} \right)_0^{\pi /2}$

$ = \frac{{{\pi ^2}}}{8}$, $ [ \because \int_{\,0}^{\,\pi /2} {[\sin x]\,dx = 0} ]$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $a \in R$ and $A$ be a matrix of order $3 \times 3$ such that $\operatorname{det}(A)=-4$ and $A+I=\left[\begin{array}{lll}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{array}\right]$, where $I$ is the identity matrix of order $3 \times 3$.
If $\operatorname{det}((a+1) \operatorname{adj}((a-1) A))$ is $2^{ m } 3^{ n }, m , n \in$ $\{0,1,2, \ldots \ldots 20\}$, then $m + n$ is equal to :

If a normal drawn to the parabola ${y^2} = 4ax$ at the point $(a,\;2a)$ meets parabola again on $(a{t^2},\;2at)$, then the value of $t$ will be

If the mean of $3, 4, x, 7, 10$ is $6$, then the value of $x$ is

A ladder is resting with the wall at an angle of ${30^o}$. A man is ascending the ladder at the rate of $ 3 \,ft/sec$. His rate of approaching the wall is

Let three vectors $\overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, $\vec{b}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{c}=x \hat{i}+y \hat{j}+z \hat{k}$ from a triangle such that $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ and the area of the triangle is $5 \sqrt{6}$. if $\alpha$ is a positive real number, then $|\overrightarrow{\mathrm{c}}|^2$ is :

If $y=y(x)$ is the solution curve of the differential equation $\left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0,$ $x>2, y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

If $y = \sin px$ and ${y_n}$ is the $n^{th}$ derivative of $y$, then $\left| {\begin{array}{*{20}{c}}
y&{{y_1}}&{{y_2}}\\
{{y_3}}&{{y_4}}&{{y_5}}\\
{{y_6}}&{{y_7}}&{{y_7}}
\end{array}} \right|$ is equal to

If the remainder when $x$ is divided by $4$ is $3 ,$ then the remainder when $(2020+ x )^{2022}$ is divided by $8$ is ....... .

Let $f(x) = 15-|x -10|;\,\,x \in R.$ then the set of all values of $x,$ at which the function, $g(x) = f(f(x))$ is not differentiable, is

If $p,q,r$ are in $G.P$ and ${\tan ^{ - 1}}p$, ${\tan ^{ - 1}}q,{\tan ^{ - 1}}r$ are in $A.P.$ then $p, q, r$ are satisfies the relation