Mathematics

$$\displaystyle \int \frac{1}{\sin ^{-1}x\sqrt{\left ( 1-x^{2} \right )}}dx.$$


ANSWER

$$\displaystyle\log \sin^{-1}x.$$


SOLUTION
Let $$ \displaystyle I= \int  \frac { 1 }{ \sin ^{ -1 } x\sqrt { \left( 1-x^{ 2 } \right)  }  } dx$$
Put $$ \displaystyle \sin ^{ -1 } x=t\Rightarrow \frac { 1 }{ \sqrt { \left( 1-x^{ 2 } \right)  }  } dx=dt$$
$$ \displaystyle I=\int { \frac { dt }{ t }  } =\log { t } +c=\log { \sin ^{ -1 } x } +c$$
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Single Correct Medium Published on 17th 09, 2020
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