Mathematics

$$\int {\frac{{{{\sin }^{ - 1}}x}}{{{{\left( {1 - {x^2}} \right)}^{\frac{3}{2}}}}}dx} $$


ANSWER

$$\frac{{x\left( {{{\sin }^{ - 1}}x} \right)}}{{\sqrt {1 - {x^2}} }} + \frac{1}{2}\log \left| {\left( {1 - {x^2}} \right)} \right| + C.$$


SOLUTION
$$\begin{array}{l} Put\, x=\sin  t\, \, so,\, that\, dx={ { costdt } }\, and\, t=si{ n^{ -1 } }x \\ \therefore \int { \frac { { { { \sin   }^{ -1 } }x } }{ { { { \left( { 1-{ x^{ 2 } } } \right)  }^{ \frac { 3 }{ 2 }  } } } } dx=\int { \frac { { t\cos  t } }{ { { { \left( { 1-{ { \sin   }^{ 2 } }t } \right)  }^{ \frac { 3 }{ 2 }  } } } } dt } =\int { \frac { { t{ { cost } } } }{ { { { \cos   }^{ 3 } }t } } dt }  }  \\ =\int { t{ { \sec   }^{ 2 } }tdt }  \\ =t.\left( { \tan  t } \right) -\int { 1.\tan  tdt }  \\ =t\left( { \tan  t } \right) +\log  \left| { \cos  t } \right| +C \\ ={ \sin ^{ -1 }  }x\frac { x }{ { \sqrt { 1-{ x^{ 2 } } }  } } +\log  \left| { \sqrt { 1-{ x^{ 2 } } }  } \right| +C \\ =\frac { { x\left( { { { \sin   }^{ -1 } }x } \right)  } }{ { \sqrt { 1-{ x^{ 2 } } }  } } +\frac { 1 }{ 2 } \log  \left| { \left( { 1-{ x^{ 2 } } } \right)  } \right| +C. \\ hence,\, the\, \, option\, \, A\, is\, correct\, \, answer. \end{array}$$
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