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

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

What is the length of the longer diagonal of the parallelogram constructed on $5\vec{\text{a}}+2\vec{\text{b}}$ and $\vec{\text{a}}-3\vec{\text{b}}$ if it is given that $|\vec{\text{a}}|=2\sqrt{2},\big|\vec{\text{b}}\big|=3$ and the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{\pi}{4}$?

$15$
$\sqrt{113}$
$\sqrt{593}$
$\sqrt{369}$

The degree and order of differential equation $y^{\prime \prime 2}+\log \left(y^{\prime}\right)=x^5$ respectively are:

If the lines $\text{ x - }\frac{2}{1} =\text{y}-\frac{2}{1} =\text{z}-\frac{4}{\text{k}} $ and $\text{x}-\frac{1}{\text{k}} = \text{y}-\frac{4}{2} = \text{z}-\frac{5}{1} $ are coplanar, then k can have:

If $\left[\begin{array}{ll}x+y & 2 x+z \\ x-y & 2 z+w\end{array}\right]=\left[\begin{array}{cc}4 & 7 \\ 0 & 10\end{array}\right]$, then the values of $x, y, z$ and $w$ respectively are

The ratio of the rate of flow of water in pipes varies inversely as the square of the radius of the pipes. What is the ratio of the rates of flow in two pipes diameters 2cm and 4cm?

1 : 6
1 : 4
1 : 2
3 : 1

If $y=y(x)$ and it follows the relation $4x{e^{xy}} = y + 5{\sin ^2}x$ ,then $y'(0)$ is equal to

If $G $ and $G' $ be the centroids of the triangles $ABC $ and $A'B'C'$ respectively, then $\overrightarrow {AA} ' + \overrightarrow {BB'} + \overrightarrow {CC} ' = $

Choose the correct answer from the given four options.
Let F = 3x - 4y be the objective function.
Minimum value of F is:

Consider the function $f (x) = x^3 - 8x^2 + 20x -13$
Number of positive integers $x$ for which $f (x)$ is a prime number, is

$(2a + 3b) \times (5a + 7b) = $