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

MCQ

$\int_{\,0}^{\,\pi /2} {\sin 2x\log \tan x\,dx} $ is equal to

A
$\pi $
B
$\pi /2$
✓
$0$
D
$2\pi $

✓

Answer

Correct option: C.

$0$

c
(c) $I = \int_0^{\pi /2} {\,\,\sin 2x\log \tan x\,\,dx} $,

$I = \int_0^{\pi /2} {\sin 2\,\left( {\frac{\pi }{2} - x} \right)\log \tan \left( {\frac{\pi }{2} - x} \right)\,\,dx} $,

$[\because \int_{0}^{a}{f\,(x)\,dx=\int_{0}^{a}{f\,(a-x)\,dx]}}$

$ = \int_0^{\pi /2} {\,\,\,\,\,\sin 2x\log \cot x\,\,dx} $

$ = - \int_0^{\pi /2} {\,\,\,\,\,\sin 2x\log \tan x\,\,dx} $

$\therefore I = - I\,\,==> 2I = 0$ $ \Rightarrow I = 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

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

If $y=\tan ^{-1}\left(\sec x^{3}-\tan x^{3}\right) \cdot \frac{\pi}{2} < x^{3} < \frac{3 \pi}{2}$, then

Let $f:R \to R$ be a function defined by $f(x) = \max \,(x,\,{x^3}).$ The set of all points where $f(x)$ is not differentiable is

If $\omega $ is cube root of unity, then root of the equation $\left| {\begin{array}{*{20}{c}}
{x + 2}&\omega &{{\omega ^2}} \\
\omega &{x + 1 + {\omega ^2}}&1 \\
{{\omega ^2}}&1&{x + 1 + \omega }
\end{array}} \right| = 0$ is

Let mininmm $m$ , $(m\in Z^+)$ is define as power of a square matrix $'A'$ such that $A^m = I$ . If $A^5 = I$ and $ABA^{-1} = B^2$ . then power of matrix $B$ is between

An integrating factor of the differential equation $(1 - {x^2})\frac{{dy}}{{dx}} - xy = 1,$ is

A unit vector which is perpendicular to the vector $2\hat i - \hat j + 2\hat k$ and is coplanar with the vectors $\hat i + \hat j - \hat k$ and $2\hat i + 2\hat j - \hat k$ is

The points of discontinuity of the function $\text{f(x)}=\begin{cases}2\sqrt{\text{x}},&0\leq\text{x}\leq1\\4-2\text{x},&1<\text{x}<\frac{5}{2}\\2\text{x}-7,&\frac{5}{2}\leq\text{x}\leq4\end{cases}$ is (are):

$\text{x}=1,\text{x}=\frac{5}{2}$
$\text{x}=\frac{5}{2}$
$\text{x}=1,\frac{5}{2},4$
$\text{x}=0,4$

If $\tan^{-1}(\cot\theta)=2\theta,$ then $\theta=$

$\pm\frac{\pi}{3}$
$\pm\frac{\pi}{4}$
$\pm\frac{\pi}{6}$
$\text{none of these}$

Let $S$ be the set of all real numbers and let $R$ be a relation on $S$ defined by $a R b \Leftrightarrow a^2+b^2=1$. Then, $R$ is

$\int_{}^{} {\frac{{dx}}{{x({x^n} + 1)}} = } $