Mathematics

Integrate $\displaystyle\int _{ 0 }^{ 1 }{ \sin ^{ -1 }{ \left( \dfrac { 2x }{ 1+{ x }^{ 2 } } \right) dx } }$

SOLUTION
Let,
$I = \int_0^1 {si{n^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right)dx}$
Let $x=tan{\theta}$
$dx=sec^2{\theta}d{\theta}$
When,  $x=0, {\theta}=0$ & $x=1, {\theta}=\frac{\pi}{4}$
Now,
$I = \int_0^{\frac{\pi }{4}} {2\theta .{{\sec }^2}\theta .d\theta }$
Integrating by parts , Taking $sec^2{\theta}$ as a first function.
$=\dfrac{\pi}{2}-\log 2$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

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