_ ,0 ^ , 1 + 2x 2 ,dx is equal to - Vidyadip

Question

$\int_{\,0}^{\,\pi } {\sqrt {\frac{{1 + \cos 2x}}{2}} \,dx} $ is equal to

✓

Answer

b
(b) $I = \int_0^\pi {\sqrt {\frac{{1 + \cos 2x}}{2}} dx = \int_0^{\,\pi } {|\cos x|\,dx} } $

$I = \int_{\,0}^{\,\pi /2} {\cos x\,dx} - \int_{\,\pi /2}^{\,\pi } {\cos x\,dx} $

$= [\sin x]_0^{\pi /2} - [\sin x]_{\pi /2}^\pi $

$I = \left[ {\sin \frac{\pi }{2} - \sin 0} \right] - \left[ {\sin \pi - \sin \frac{\pi }{2}} \right] $

$=1+ 1 = 2.$

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

For each real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$, and let $\{ x \}= x -[ x ]$. Then the smallest positive integer $M$ for which $\int_1^M\{x\}^{[x]} d x > 1$ is

If three conterminous edges of a parallelopiped are represented by $a - b,\,\,b - c$ and $c - a$, then its volume is

Let for $x \in R , S_0( x )= x$,$S _{ k }( x )= C _{ k } x + k \int _0^{ x } S _{ k -1}(t) d t$, where $C _0=1, C _{ k }=1-\int_0^1 S _{ k -1}( x ) dx , k =1,2,3 \ldots$. Then $S _2(3)+6 C _3$ is equal to $...........$.

For $k \in N$, if the sum of the series $1+\frac{4}{ k }+\frac{8}{ k ^2}+\frac{13}{ k ^3}+\frac{19}{ k ^4}+\ldots$ is 10 , then the value of $k$ is

The value of $\left| {\,\begin{array}{*{20}{c}}{265}&{240}&{219}\\{240}&{225}&{198}\\{219}&{198}&{181}\end{array}\,} \right|$ is equal to

If the shortest distance between the lines $\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}, $ then the largest possible value of $|\lambda|$ is equal $..........$

The number of common tangents to the circles ${x^2} + {y^2} - x = 0,\,{x^2} + {y^2} + x = 0$ is

If the matrix $\left[ {\begin{array}{*{20}{c}}1&3&{\lambda + 2}\\2&4&8\\3&5&{10}\end{array}} \right]$ is singular, then $\lambda = $

If ${{2x + 3} \over {(x + 1)(x - 3)}} = {a \over {x + 1}} + {b \over {(x - 3)}}$, then $a + b$

Given the relation $R = \{(1, 2), (2, 3)\}$ on the set $A = {1, 2, 3}$, the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is