Mathematics

# $\displaystyle\int _{ 0 }^{ 1 }{ \dfrac { 2\sin ^{ -1 }{ \dfrac { x }{ 2 } } }{ x } } dx$ is equal to

$\displaystyle\int _{ 0 }^{ \pi /6 }{ \cfrac { 2x }{ \tan { x } } } dx$

##### SOLUTION
$I=\displaystyle\int^1_0\dfrac{2\sin^{-1}\dfrac{x}{2}}{x}dx$
put $\sin^{-1}\dfrac{x}{2}=t$
$\Rightarrow \dfrac{x}{2}=\sin t$
$\Rightarrow x=2\sin t$
$\therefore dx=2\cos t dt$
As $x+0, t\rightarrow 0$
$x\rightarrow 1, t\rightarrow \dfrac{\pi}{6}$
$=\displaystyle\int^{\dfrac{\pi}{6}}_0\dfrac{2t}{2\sin t}2\cos tdt$
$=\displaystyle\int^{\dfrac{\pi}{6}}_0\dfrac{2t}{\tan t}dt$
$=\displaystyle\int^{\dfrac{\pi}{6}}_0\dfrac{2x}{\tan x}dx$.

Its FREE, you're just one step away

Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Solve:
$\frac{1}{2}\int {\frac{{2x}}{{{{\left( {1 + x} \right)}^2}}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate the function    $\displaystyle \frac {1}{1-\tan x}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate $\displaystyle \int { \dfrac { \cos ^{ 4 }{ x } }{ \sin ^{ 3 }{ x } \left( \sin ^{ 5 }{ x } +\cos ^{ 5 }{ x } \right) ^{ 3/5 } } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve :
$I = \displaystyle\int \dfrac {x+9}{x^2+5} dx$

$\int{\left( 3x+2 \right)\sqrt{10-4x-3{{x}^{2}}}}dx$