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
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