The integral _ /6 ^ /4 dx ,2x , ( ^5 ,x + ^5 ,x ) equals - Vidyadip

MCQ

The integral $\int_{\pi /6}^{\pi /4} {\frac{{dx}}{{\sin \,2x\,\left( {{{\tan }^5}\,x + {{\cot }^5}\,x} \right)}}} $ equals

A
$\frac{1}{{20}}\,{\tan ^{ - 1}}\,\left( {\frac{1}{{9\sqrt 3 }}} \right)$
✓
$\frac{1}{{10}}\,\left( {\frac{\pi }{4} - {{\tan }^{ - 1}}\,\left( {\frac{1}{{9\sqrt {1\sqrt 3 } }}} \right)} \right)$
C
$\frac{\pi }{{40}}$
D
$\frac{1}{5}\,\left( {\frac{\pi }{4} - {{\tan }^{ - 1}}\,\left( {\frac{1}{{3\sqrt 3 }}} \right)} \right)$

✓

Answer

Correct option: B.

$\frac{1}{{10}}\,\left( {\frac{\pi }{4} - {{\tan }^{ - 1}}\,\left( {\frac{1}{{9\sqrt {1\sqrt 3 } }}} \right)} \right)$

b
$I = \int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\frac{{{{\sec }^2}xdx}}{{2\tan x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}} $

$\text { Put } \tan x=t$

$ = \int\limits_{\frac{1}{{\sqrt 3 }}}^1 {\frac{{dt}}{{2t\left( {{t^5} + \frac{1}{{{t^5}}}} \right)}}} $

$ = \int\limits_{\frac{1}{{\sqrt 3 }}}^1 {\frac{{{t^4}dt}}{{2\left( {{t^{10}} + 1} \right)}}} $

$\text { Put } t^{5}=y$

$\left.\mathrm{I}=\frac{1}{10} \tan ^{-1}(\mathrm{y})\right]_{3}^{1} \frac{\mathrm{s}}{2}$

$=\frac{1}{10}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{3^{5 / 2}}\right)\right)$

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

${\sin ^2}\frac{\pi }{8} + {\sin ^2}\frac{{3\pi }}{8} + {\sin ^2}\frac{{5\pi }}{8} + {\sin ^2}\frac{{7\pi }}{8} = $

Let ${a_1},{a_2},.......,{a_{30}}$ be an $A.P.$, $S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{2i - 1}}} $.If ${a_5} = 27\,a$ and $S - 2T = 75$ , then $a_{10}$ is equal to

Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\overrightarrow{ b } \cdot \overrightarrow{ c }$ is

$\sqrt 2 \smallint \frac{{sinx\;dx}}{{{\rm{sin}}\left( {x - \frac{\pi }{4}} \right)}} = $

If $\omega $ is a cube root of unity and $\Delta = \left| {\begin{array}{*{20}{c}}1&{2\omega }\\\omega &{{\omega ^2}}\end{array}} \right|$, then ${\Delta ^2}$ is equal to

The number of positive integral solutions of $a\,b\,c = 30$ is

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, ……$ be $50\,n\, + \,\frac{{n\,(n\, - 7)}}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.,$ then the ordered pair $(d,a_{50})$ is equal to

The sum of infinite terms of a $G.P.$ is $x$ and on squaring the each term of it, the sum will be $y$, then the common ratio of this series is

$\int {\frac{{{x^2}}}{{{x^2} + 4}}\,\,dx} $ equals to

The number of ways in which $5$ boys and $3$ girls can be seated in a row so that each girl in between two boys